Theorem remulscl 28229

Description: The surreal reals are closed under multiplication. Part of theorem 13(ii) of [Conway] p. 24. (Contributed by Scott Fenton, 16-Apr-2025.)

Assertion

Ref	Expression
remulscl	⊢ ((𝐴 ∈ ℝs ∧ 𝐵 ∈ ℝs) → (𝐴 ·s 𝐵) ∈ ℝs)

Proof of Theorem remulscl

Dummy variables 𝑥 𝑦 𝑧 𝑛 𝑚 𝑝 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulscl 28033	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) ∈ No )
2	1	adantr 480	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})))) → (𝐴 ·s 𝐵) ∈ No )
3		remulscllem2 28228	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → ∃𝑝 ∈ ℕs (( -us ‘𝑝) <s (𝐴 ·s 𝐵) ∧ (𝐴 ·s 𝐵) <s 𝑝))
4	3	expr 456	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)) → ∃𝑝 ∈ ℕs (( -us ‘𝑝) <s (𝐴 ·s 𝐵) ∧ (𝐴 ·s 𝐵) <s 𝑝)))
5	4	rexlimdvva 3208	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)) → ∃𝑝 ∈ ℕs (( -us ‘𝑝) <s (𝐴 ·s 𝐵) ∧ (𝐴 ·s 𝐵) <s 𝑝)))
6		simpl 482	. . . . . . 7 ⊢ ((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) → ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛))
7		simpl 482	. . . . . . 7 ⊢ ((∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})) → ∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚))
8	6, 7	anim12i 612	. . . . . 6 ⊢ (((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ ∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))
9		reeanv 3223	. . . . . 6 ⊢ (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)) ↔ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ ∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))
10	8, 9	sylibr 233	. . . . 5 ⊢ (((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))
11	5, 10	impel 505	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})))) → ∃𝑝 ∈ ℕs (( -us ‘𝑝) <s (𝐴 ·s 𝐵) ∧ (𝐴 ·s 𝐵) <s 𝑝))
12		simpr 484	. . . . . 6 ⊢ ((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) → 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))
13		simpr 484	. . . . . 6 ⊢ ((∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})) → 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))
14	12, 13	anim12i 612	. . . . 5 ⊢ (((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})))
15		recut 28223	. . . . . . . . 9 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})
16	15	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})
17	16	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})
18		recut 28223	. . . . . . . 8 ⊢ (𝐵 ∈ No → {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} <<s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})
19	18	ad2antlr 726	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} <<s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})
20		simprl 770	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))
21		simprr 772	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))
22	17, 19, 20, 21	mulsunif2 28069	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → (𝐴 ·s 𝐵) = (({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))}) |s ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))})))
23		r19.41v 3185	. . . . . . . . . . . . . 14 ⊢ (∃𝑛 ∈ ℕs (𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))) ↔ (∃𝑛 ∈ ℕs 𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))))
24	23	exbii 1843	. . . . . . . . . . . . 13 ⊢ (∃𝑡∃𝑛 ∈ ℕs (𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑡(∃𝑛 ∈ ℕs 𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))))
25		rexcom4 3282	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑡∃𝑛 ∈ ℕs (𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))))
26		eqeq1 2732	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑡 = (𝐴 -s ( 1s /su 𝑛))))
27	26	rexbidv 3175	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑡 = (𝐴 -s ( 1s /su 𝑛))))
28	27	rexab 3689	. . . . . . . . . . . . 13 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢))) ↔ ∃𝑡(∃𝑛 ∈ ℕs 𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))))
29	24, 25, 28	3bitr4ri 304	. . . . . . . . . . . 12 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢))) ↔ ∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))))
30		ovex 7453	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 -s ( 1s /su 𝑛)) ∈ V
31		oveq2 7428	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → (𝐴 -s 𝑡) = (𝐴 -s (𝐴 -s ( 1s /su 𝑛))))
32	31	oveq1d 7435	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)) = ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢)))
33	32	oveq2d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢))) = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢))))
34	33	eqeq2d 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → (𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢))) ↔ 𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢)))))
35	34	rexbidv 3175	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → (∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢))) ↔ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢)))))
36	30, 35	ceqsexv 3523	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢))))
37		r19.41v 3185	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑚 ∈ ℕs (𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢)))) ↔ (∃𝑚 ∈ ℕs 𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢)))))
38	37	exbii 1843	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑢∃𝑚 ∈ ℕs (𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑢(∃𝑚 ∈ ℕs 𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢)))))
39		rexcom4 3282	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑚 ∈ ℕs ∃𝑢(𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑢∃𝑚 ∈ ℕs (𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢)))))
40		eqeq1 2732	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑢 → (𝑦 = (𝐵 -s ( 1s /su 𝑚)) ↔ 𝑢 = (𝐵 -s ( 1s /su 𝑚))))
41	40	rexbidv 3175	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑢 → (∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚)) ↔ ∃𝑚 ∈ ℕs 𝑢 = (𝐵 -s ( 1s /su 𝑚))))
42	41	rexab 3689	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢))) ↔ ∃𝑢(∃𝑚 ∈ ℕs 𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢)))))
43	38, 39, 42	3bitr4ri 304	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢))) ↔ ∃𝑚 ∈ ℕs ∃𝑢(𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢)))))
44	36, 43	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑚 ∈ ℕs ∃𝑢(𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢)))))
45		ovex 7453	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 -s ( 1s /su 𝑚)) ∈ V
46		oveq2 7428	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = (𝐵 -s ( 1s /su 𝑚)) → (𝐵 -s 𝑢) = (𝐵 -s (𝐵 -s ( 1s /su 𝑚))))
47	46	oveq2d 7436	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝐵 -s ( 1s /su 𝑚)) → ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢)) = ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s (𝐵 -s ( 1s /su 𝑚)))))
48	47	oveq2d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = (𝐵 -s ( 1s /su 𝑚)) → ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢))) = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s (𝐵 -s ( 1s /su 𝑚))))))
49	48	eqeq2d 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = (𝐵 -s ( 1s /su 𝑚)) → (𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢))) ↔ 𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s (𝐵 -s ( 1s /su 𝑚)))))))
50	45, 49	ceqsexv 3523	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑢(𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢)))) ↔ 𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s (𝐵 -s ( 1s /su 𝑚))))))
51		simplll 774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → 𝐴 ∈ No )
52		1sno 27759	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1s ∈ No
53	52	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → 1s ∈ No )
54		nnsno 28195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
55	54	ad2antlr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → 𝑛 ∈ No )
56		nnne0s 28204	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
57	56	ad2antlr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → 𝑛 ≠ 0s )
58	53, 55, 57	divscld 28121	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
59	51, 58	nncansd 28002	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → (𝐴 -s (𝐴 -s ( 1s /su 𝑛))) = ( 1s /su 𝑛))
60		simpllr 775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → 𝐵 ∈ No )
61		nnsno 28195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ No )
62	61	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → 𝑚 ∈ No )
63		nnne0s 28204	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕs → 𝑚 ≠ 0s )
64	63	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → 𝑚 ≠ 0s )
65	53, 62, 64	divscld 28121	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → ( 1s /su 𝑚) ∈ No )
66	60, 65	nncansd 28002	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → (𝐵 -s (𝐵 -s ( 1s /su 𝑚))) = ( 1s /su 𝑚))
67	59, 66	oveq12d 7438	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s (𝐵 -s ( 1s /su 𝑚)))) = (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))
68	67	oveq2d 7436	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s (𝐵 -s ( 1s /su 𝑚))))) = ((𝐴 ·s 𝐵) -s (( 1s /su 𝑛) ·s ( 1s /su 𝑚))))
69	68	eqeq2d 2739	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → (𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s (𝐵 -s ( 1s /su 𝑚))))) ↔ 𝑧 = ((𝐴 ·s 𝐵) -s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
70	50, 69	bitrid 283	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → (∃𝑢(𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢)))) ↔ 𝑧 = ((𝐴 ·s 𝐵) -s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
71	70	rexbidva 3173	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → (∃𝑚 ∈ ℕs ∃𝑢(𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
72	44, 71	bitrid 283	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → (∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
73	72	rexbidva 3173	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
74		remulscllem1 28227	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s (( 1s /su 𝑛) ·s ( 1s /su 𝑚))) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝)))
75	73, 74	bitrdi 287	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))))
76	29, 75	bitrid 283	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢))) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))))
77	76	abbidv 2797	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))} = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))})
78		r19.41v 3185	. . . . . . . . . . . . . 14 ⊢ (∃𝑛 ∈ ℕs (𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))) ↔ (∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))))
79	78	exbii 1843	. . . . . . . . . . . . 13 ⊢ (∃𝑡∃𝑛 ∈ ℕs (𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑡(∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))))
80		rexcom4 3282	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑡∃𝑛 ∈ ℕs (𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))))
81		eqeq1 2732	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑡 = (𝐴 +s ( 1s /su 𝑛))))
82	81	rexbidv 3175	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛))))
83	82	rexab 3689	. . . . . . . . . . . . 13 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵))) ↔ ∃𝑡(∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))))
84	79, 80, 83	3bitr4ri 304	. . . . . . . . . . . 12 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵))) ↔ ∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))))
85		ovex 7453	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 +s ( 1s /su 𝑛)) ∈ V
86		oveq1 7427	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → (𝑡 -s 𝐴) = ((𝐴 +s ( 1s /su 𝑛)) -s 𝐴))
87	86	oveq1d 7435	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)) = (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵)))
88	87	oveq2d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵))) = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵))))
89	88	eqeq2d 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → (𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵))) ↔ 𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵)))))
90	89	rexbidv 3175	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → (∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵))) ↔ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵)))))
91	85, 90	ceqsexv 3523	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵))))
92		r19.41v 3185	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑚 ∈ ℕs (𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵)))) ↔ (∃𝑚 ∈ ℕs 𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵)))))
93	92	exbii 1843	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑢∃𝑚 ∈ ℕs (𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑢(∃𝑚 ∈ ℕs 𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵)))))
94		rexcom4 3282	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑚 ∈ ℕs ∃𝑢(𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑢∃𝑚 ∈ ℕs (𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵)))))
95		eqeq1 2732	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑢 → (𝑦 = (𝐵 +s ( 1s /su 𝑚)) ↔ 𝑢 = (𝐵 +s ( 1s /su 𝑚))))
96	95	rexbidv 3175	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑢 → (∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚)) ↔ ∃𝑚 ∈ ℕs 𝑢 = (𝐵 +s ( 1s /su 𝑚))))
97	96	rexab 3689	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵))) ↔ ∃𝑢(∃𝑚 ∈ ℕs 𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵)))))
98	93, 94, 97	3bitr4ri 304	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵))) ↔ ∃𝑚 ∈ ℕs ∃𝑢(𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵)))))
99	91, 98	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑚 ∈ ℕs ∃𝑢(𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵)))))
100		ovex 7453	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 +s ( 1s /su 𝑚)) ∈ V
101		oveq1 7427	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = (𝐵 +s ( 1s /su 𝑚)) → (𝑢 -s 𝐵) = ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵))
102	101	oveq2d 7436	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝐵 +s ( 1s /su 𝑚)) → (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵)) = (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵)))
103	102	oveq2d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = (𝐵 +s ( 1s /su 𝑚)) → ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵))) = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵))))
104	103	eqeq2d 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = (𝐵 +s ( 1s /su 𝑚)) → (𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵))) ↔ 𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵)))))
105	100, 104	ceqsexv 3523	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑢(𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵)))) ↔ 𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵))))
106		pncan2s 27981	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No ) → ((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) = ( 1s /su 𝑛))
107	51, 58, 106	syl2anc 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → ((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) = ( 1s /su 𝑛))
108		pncan2s 27981	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 ∈ No ∧ ( 1s /su 𝑚) ∈ No ) → ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵) = ( 1s /su 𝑚))
109	60, 65, 108	syl2anc 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵) = ( 1s /su 𝑚))
110	107, 109	oveq12d 7438	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵)) = (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))
111	110	oveq2d 7436	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵))) = ((𝐴 ·s 𝐵) -s (( 1s /su 𝑛) ·s ( 1s /su 𝑚))))
112	111	eqeq2d 2739	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → (𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵))) ↔ 𝑧 = ((𝐴 ·s 𝐵) -s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
113	105, 112	bitrid 283	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → (∃𝑢(𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵)))) ↔ 𝑧 = ((𝐴 ·s 𝐵) -s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
114	113	rexbidva 3173	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → (∃𝑚 ∈ ℕs ∃𝑢(𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) -s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
115	99, 114	bitrid 283	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → (∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
116	115	rexbidva 3173	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
117	116, 74	bitrdi 287	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))))
118	84, 117	bitrid 283	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵))) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))))
119	118	abbidv 2797	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))} = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))})
120	77, 119	uneq12d 4163	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))}) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))} ∪ {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))}))
121		unidm 4151	. . . . . . . . 9 ⊢ ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))} ∪ {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))}) = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))}
122	120, 121	eqtrdi 2784	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))}) = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))})
123		r19.41v 3185	. . . . . . . . . . . . . 14 ⊢ (∃𝑛 ∈ ℕs (𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))) ↔ (∃𝑛 ∈ ℕs 𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))))
124	123	exbii 1843	. . . . . . . . . . . . 13 ⊢ (∃𝑡∃𝑛 ∈ ℕs (𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑡(∃𝑛 ∈ ℕs 𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))))
125		rexcom4 3282	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑡∃𝑛 ∈ ℕs (𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))))
126	27	rexab 3689	. . . . . . . . . . . . 13 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵))) ↔ ∃𝑡(∃𝑛 ∈ ℕs 𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))))
127	124, 125, 126	3bitr4ri 304	. . . . . . . . . . . 12 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵))) ↔ ∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))))
128	31	oveq1d 7435	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)) = ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵)))
129	128	oveq2d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵))) = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵))))
130	129	eqeq2d 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → (𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵))) ↔ 𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵)))))
131	130	rexbidv 3175	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → (∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵))) ↔ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵)))))
132	30, 131	ceqsexv 3523	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵))))
133		r19.41v 3185	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑚 ∈ ℕs (𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵)))) ↔ (∃𝑚 ∈ ℕs 𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵)))))
134	133	exbii 1843	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑢∃𝑚 ∈ ℕs (𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑢(∃𝑚 ∈ ℕs 𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵)))))
135		rexcom4 3282	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑚 ∈ ℕs ∃𝑢(𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑢∃𝑚 ∈ ℕs (𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵)))))
136	96	rexab 3689	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵))) ↔ ∃𝑢(∃𝑚 ∈ ℕs 𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵)))))
137	134, 135, 136	3bitr4ri 304	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵))) ↔ ∃𝑚 ∈ ℕs ∃𝑢(𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵)))))
138	132, 137	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑚 ∈ ℕs ∃𝑢(𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵)))))
139	101	oveq2d 7436	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝐵 +s ( 1s /su 𝑚)) → ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵)) = ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵)))
140	139	oveq2d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = (𝐵 +s ( 1s /su 𝑚)) → ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵))) = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵))))
141	140	eqeq2d 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = (𝐵 +s ( 1s /su 𝑚)) → (𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵))) ↔ 𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵)))))
142	100, 141	ceqsexv 3523	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑢(𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵)))) ↔ 𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵))))
143	59, 109	oveq12d 7438	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵)) = (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))
144	143	oveq2d 7436	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵))) = ((𝐴 ·s 𝐵) +s (( 1s /su 𝑛) ·s ( 1s /su 𝑚))))
145	144	eqeq2d 2739	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → (𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s ((𝐵 +s ( 1s /su 𝑚)) -s 𝐵))) ↔ 𝑧 = ((𝐴 ·s 𝐵) +s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
146	142, 145	bitrid 283	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → (∃𝑢(𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵)))) ↔ 𝑧 = ((𝐴 ·s 𝐵) +s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
147	146	rexbidva 3173	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → (∃𝑚 ∈ ℕs ∃𝑢(𝑢 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s (𝐴 -s ( 1s /su 𝑛))) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
148	138, 147	bitrid 283	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → (∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
149	148	rexbidva 3173	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
150		remulscllem1 28227	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s (( 1s /su 𝑛) ·s ( 1s /su 𝑚))) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝)))
151	149, 150	bitrdi 287	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))))
152	127, 151	bitrid 283	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵))) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))))
153	152	abbidv 2797	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))} = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))})
154		r19.41v 3185	. . . . . . . . . . . . . 14 ⊢ (∃𝑛 ∈ ℕs (𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))) ↔ (∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))))
155	154	exbii 1843	. . . . . . . . . . . . 13 ⊢ (∃𝑡∃𝑛 ∈ ℕs (𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑡(∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))))
156		rexcom4 3282	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑡∃𝑛 ∈ ℕs (𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))))
157	82	rexab 3689	. . . . . . . . . . . . 13 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢))) ↔ ∃𝑡(∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))))
158	155, 156, 157	3bitr4ri 304	. . . . . . . . . . . 12 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢))) ↔ ∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))))
159	86	oveq1d 7435	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)) = (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢)))
160	159	oveq2d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢))) = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢))))
161	160	eqeq2d 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → (𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢))) ↔ 𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢)))))
162	161	rexbidv 3175	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → (∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢))) ↔ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢)))))
163	85, 162	ceqsexv 3523	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢))))
164		r19.41v 3185	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑚 ∈ ℕs (𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢)))) ↔ (∃𝑚 ∈ ℕs 𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢)))))
165	164	exbii 1843	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑢∃𝑚 ∈ ℕs (𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑢(∃𝑚 ∈ ℕs 𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢)))))
166		rexcom4 3282	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑚 ∈ ℕs ∃𝑢(𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑢∃𝑚 ∈ ℕs (𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢)))))
167	41	rexab 3689	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢))) ↔ ∃𝑢(∃𝑚 ∈ ℕs 𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢)))))
168	165, 166, 167	3bitr4ri 304	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢))) ↔ ∃𝑚 ∈ ℕs ∃𝑢(𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢)))))
169	163, 168	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑚 ∈ ℕs ∃𝑢(𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢)))))
170	46	oveq2d 7436	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝐵 -s ( 1s /su 𝑚)) → (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢)) = (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s (𝐵 -s ( 1s /su 𝑚)))))
171	170	oveq2d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = (𝐵 -s ( 1s /su 𝑚)) → ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢))) = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s (𝐵 -s ( 1s /su 𝑚))))))
172	171	eqeq2d 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = (𝐵 -s ( 1s /su 𝑚)) → (𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢))) ↔ 𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s (𝐵 -s ( 1s /su 𝑚)))))))
173	45, 172	ceqsexv 3523	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑢(𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢)))) ↔ 𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s (𝐵 -s ( 1s /su 𝑚))))))
174	107, 66	oveq12d 7438	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s (𝐵 -s ( 1s /su 𝑚)))) = (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))
175	174	oveq2d 7436	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s (𝐵 -s ( 1s /su 𝑚))))) = ((𝐴 ·s 𝐵) +s (( 1s /su 𝑛) ·s ( 1s /su 𝑚))))
176	175	eqeq2d 2739	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → (𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s (𝐵 -s ( 1s /su 𝑚))))) ↔ 𝑧 = ((𝐴 ·s 𝐵) +s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
177	173, 176	bitrid 283	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → (∃𝑢(𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢)))) ↔ 𝑧 = ((𝐴 ·s 𝐵) +s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
178	177	rexbidva 3173	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → (∃𝑚 ∈ ℕs ∃𝑢(𝑢 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = ((𝐴 ·s 𝐵) +s (((𝐴 +s ( 1s /su 𝑛)) -s 𝐴) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
179	169, 178	bitrid 283	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → (∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
180	179	rexbidva 3173	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s (( 1s /su 𝑛) ·s ( 1s /su 𝑚)))))
181	180, 150	bitrdi 287	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ ∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))))
182	158, 181	bitrid 283	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢))) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))))
183	182	abbidv 2797	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))} = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))})
184	153, 183	uneq12d 4163	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))}) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))} ∪ {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))}))
185		unidm 4151	. . . . . . . . 9 ⊢ ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))} ∪ {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))}) = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))}
186	184, 185	eqtrdi 2784	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))}) = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))})
187	122, 186	oveq12d 7438	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))}) |s ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))})) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))} |s {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))}))
188	187	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → (({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑡) ·s (𝐵 -s 𝑢)))} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) -s ((𝑡 -s 𝐴) ·s (𝑢 -s 𝐵)))}) |s ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}∃𝑢 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = ((𝐴 ·s 𝐵) +s ((𝑡 -s 𝐴) ·s (𝐵 -s 𝑢)))})) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))} |s {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))}))
189	22, 188	eqtrd 2768	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → (𝐴 ·s 𝐵) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))} |s {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))}))
190	14, 189	sylan2 592	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})))) → (𝐴 ·s 𝐵) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))} |s {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))}))
191	2, 11, 190	jca32 515	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})))) → ((𝐴 ·s 𝐵) ∈ No ∧ (∃𝑝 ∈ ℕs (( -us ‘𝑝) <s (𝐴 ·s 𝐵) ∧ (𝐴 ·s 𝐵) <s 𝑝) ∧ (𝐴 ·s 𝐵) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))} |s {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))}))))
192	191	an4s 659	. 2 ⊢ (((𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))) ∧ (𝐵 ∈ No ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})))) → ((𝐴 ·s 𝐵) ∈ No ∧ (∃𝑝 ∈ ℕs (( -us ‘𝑝) <s (𝐴 ·s 𝐵) ∧ (𝐴 ·s 𝐵) <s 𝑝) ∧ (𝐴 ·s 𝐵) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))} |s {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))}))))
193		elreno 28222	. . 3 ⊢ (𝐴 ∈ ℝs ↔ (𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))))
194		elreno 28222	. . 3 ⊢ (𝐵 ∈ ℝs ↔ (𝐵 ∈ No ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))))
195	193, 194	anbi12i 627	. 2 ⊢ ((𝐴 ∈ ℝs ∧ 𝐵 ∈ ℝs) ↔ ((𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))) ∧ (𝐵 ∈ No ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})))))
196		elreno 28222	. 2 ⊢ ((𝐴 ·s 𝐵) ∈ ℝs ↔ ((𝐴 ·s 𝐵) ∈ No ∧ (∃𝑝 ∈ ℕs (( -us ‘𝑝) <s (𝐴 ·s 𝐵) ∧ (𝐴 ·s 𝐵) <s 𝑝) ∧ (𝐴 ·s 𝐵) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) -s ( 1s /su 𝑝))} |s {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 ·s 𝐵) +s ( 1s /su 𝑝))}))))
197	192, 195, 196	3imtr4i 292	1 ⊢ ((𝐴 ∈ ℝs ∧ 𝐵 ∈ ℝs) → (𝐴 ·s 𝐵) ∈ ℝs)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534  ∃wex 1774   ∈ wcel 2099  {cab 2705   ≠ wne 2937  ∃wrex 3067   ∪ cun 3945   class class class wbr 5148  ‘cfv 6548  (class class class)co 7420   No csur 27572   <s cslt 27573   <<s csslt 27712   |s cscut 27714   0_s c0s 27754   1_s c1s 27755   +_s cadds 27875   -u_s cnegs 27931   -_s csubs 27932   ·_s cmuls 28005   /_su cdivs 28086  ℕ_scnns 28185  ℝ_screno 28220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-dc 10469

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-ot 4638  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-2o 8487  df-oadd 8490  df-nadd 8686  df-no 27575  df-slt 27576  df-bday 27577  df-sle 27677  df-sslt 27713  df-scut 27715  df-0s 27756  df-1s 27757  df-made 27773  df-old 27774  df-left 27776  df-right 27777  df-norec 27854  df-norec2 27865  df-adds 27876  df-negs 27933  df-subs 27934  df-muls 28006  df-divs 28087  df-abss 28131  df-n0s 28186  df-nns 28187  df-reno 28221

This theorem is referenced by: (None)