Theorem remulscl 28918

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 28550	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 ·s 𝐵) ∈ No)
2	1	adantr 486	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ ((∃𝑛 ∈ ℕs (( -s‘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 (( -s‘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 28917	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -s‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -s‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → ∃𝑝 ∈ ℕs (( -s‘𝑝) <s (𝐴 ·s 𝐵) ∧ (𝐴 ·s 𝐵) <s 𝑝))
4	3	expr 462	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (((( -s‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -s‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)) → ∃𝑝 ∈ ℕs (( -s‘𝑝) <s (𝐴 ·s 𝐵) ∧ (𝐴 ·s 𝐵) <s 𝑝)))
5	4	rexlimdvva 3223	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs ((( -s‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -s‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)) → ∃𝑝 ∈ ℕs (( -s‘𝑝) <s (𝐴 ·s 𝐵) ∧ (𝐴 ·s 𝐵) <s 𝑝)))
6		simpl 488	. . . . . . 7 ⊢ ((∃𝑛 ∈ ℕs (( -s‘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 (( -s‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛))
7		simpl 488	. . . . . . 7 ⊢ ((∃𝑚 ∈ ℕs (( -s‘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 (( -s‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚))
8	6, 7	anim12i 625	. . . . . 6 ⊢ (((∃𝑛 ∈ ℕs (( -s‘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 (( -s‘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 (( -s‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ ∃𝑚 ∈ ℕs (( -s‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))
9		reeanv 3238	. . . . . 6 ⊢ (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs ((( -s‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -s‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)) ↔ (∃𝑛 ∈ ℕs (( -s‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ ∃𝑚 ∈ ℕs (( -s‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))
10	8, 9	sylibr 237	. . . . 5 ⊢ (((∃𝑛 ∈ ℕs (( -s‘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 (( -s‘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 ((( -s‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -s‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))
11	5, 10	impel 515	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ ((∃𝑛 ∈ ℕs (( -s‘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 (( -s‘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 (( -s‘𝑝) <s (𝐴 ·s 𝐵) ∧ (𝐴 ·s 𝐵) <s 𝑝))
12		simpr 490	. . . . . 6 ⊢ ((∃𝑛 ∈ ℕs (( -s‘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 490	. . . . . 6 ⊢ ((∃𝑚 ∈ ℕs (( -s‘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 625	. . . . 5 ⊢ (((∃𝑛 ∈ ℕs (( -s‘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 (( -s‘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 28910	. . . . . . . . 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 486	. . . . . . . 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 486	. . . . . . 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 28910	. . . . . . . 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 740	. . . . . . 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 783	. . . . . . 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 785	. . . . . . 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 28586	. . . . . 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 3196	. . . . . . . . . . . . . 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 1881	. . . . . . . . . . . . 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 3293	. . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → (𝑥 = (𝐴 −s ( 1s /su 𝑛)) ↔ 𝑡 = (𝐴 −s ( 1s /su 𝑛))))
27	26	rexbidv 3190	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑡 = (𝐴 −s ( 1s /su 𝑛))))
28	27	rexab 3656	. . . . . . . . . . . . 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 307	. . . . . . . . . . . 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 7476	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 −s ( 1s /su 𝑛)) ∈ V
31		oveq2 7451	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = (𝐴 −s ( 1s /su 𝑛)) → (𝐴 −s 𝑡) = (𝐴 −s (𝐴 −s ( 1s /su 𝑛))))
32	31	oveq1d 7458	. . . . . . . . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . 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 3190	. . . . . . . . . . . . . . . . 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 3502	. . . . . . . . . . . . . . . 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 3196	. . . . . . . . . . . . . . . . . 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 1881	. . . . . . . . . . . . . . . . 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 3293	. . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑢 → (𝑦 = (𝐵 −s ( 1s /su 𝑚)) ↔ 𝑢 = (𝐵 −s ( 1s /su 𝑚))))
41	40	rexbidv 3190	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑢 → (∃𝑚 ∈ ℕs 𝑦 = (𝐵 −s ( 1s /su 𝑚)) ↔ ∃𝑚 ∈ ℕs 𝑢 = (𝐵 −s ( 1s /su 𝑚))))
42	41	rexab 3656	. . . . . . . . . . . . . . . . 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 307	. . . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . . 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 7476	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 −s ( 1s /su 𝑚)) ∈ V
46		oveq2 7451	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = (𝐵 −s ( 1s /su 𝑚)) → (𝐵 −s 𝑢) = (𝐵 −s (𝐵 −s ( 1s /su 𝑚))))
47	46	oveq2d 7459	. . . . . . . . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . 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 3502	. . . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → 𝐴 ∈ No)
52		1no 28226	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1s ∈ No
53	52	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → 1s ∈ No)
54		nnno 28740	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No)
55	54	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → 𝑛 ∈ No)
56		nnne0s 28753	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
57	56	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → 𝑛 ≠ 0s )
58	53, 55, 57	divscld 28640	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → ( 1s /su 𝑛) ∈ No)
59	51, 58	nncansd 28513	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → (𝐴 −s (𝐴 −s ( 1s /su 𝑛))) = ( 1s /su 𝑛))
60		simpllr 788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → 𝐵 ∈ No)
61		nnno 28740	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ No)
62	61	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → 𝑚 ∈ No)
63		nnne0s 28753	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕs → 𝑚 ≠ 0s )
64	63	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → 𝑚 ≠ 0s )
65	53, 62, 64	divscld 28640	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → ( 1s /su 𝑚) ∈ No)
66	60, 65	nncansd 28513	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → (𝐵 −s (𝐵 −s ( 1s /su 𝑚))) = ( 1s /su 𝑚))
67	59, 66	oveq12d 7461	. . . . . . . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . 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 286	. . . . . . . . . . . . . . . 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 3188	. . . . . . . . . . . . . . 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 286	. . . . . . . . . . . . . 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 3188	. . . . . . . . . . . . 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 28916	. . . . . . . . . . . . 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 290	. . . . . . . . . . . 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 286	. . . . . . . . . . 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 2829	. . . . . . . . . 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 3196	. . . . . . . . . . . . . 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 1881	. . . . . . . . . . . . 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 3293	. . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑡 = (𝐴 +s ( 1s /su 𝑛))))
82	81	rexbidv 3190	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛))))
83	82	rexab 3656	. . . . . . . . . . . . 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 307	. . . . . . . . . . . 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 7476	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 +s ( 1s /su 𝑛)) ∈ V
86		oveq1 7450	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → (𝑡 −s 𝐴) = ((𝐴 +s ( 1s /su 𝑛)) −s 𝐴))
87	86	oveq1d 7458	. . . . . . . . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . 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 3190	. . . . . . . . . . . . . . . . 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 3502	. . . . . . . . . . . . . . . 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 3196	. . . . . . . . . . . . . . . . . 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 1881	. . . . . . . . . . . . . . . . 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 3293	. . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑢 → (𝑦 = (𝐵 +s ( 1s /su 𝑚)) ↔ 𝑢 = (𝐵 +s ( 1s /su 𝑚))))
96	95	rexbidv 3190	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑢 → (∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚)) ↔ ∃𝑚 ∈ ℕs 𝑢 = (𝐵 +s ( 1s /su 𝑚))))
97	96	rexab 3656	. . . . . . . . . . . . . . . . 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 307	. . . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . . 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 7476	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 +s ( 1s /su 𝑚)) ∈ V
101		oveq1 7450	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = (𝐵 +s ( 1s /su 𝑚)) → (𝑢 −s 𝐵) = ((𝐵 +s ( 1s /su 𝑚)) −s 𝐵))
102	101	oveq2d 7459	. . . . . . . . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . 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 3502	. . . . . . . . . . . . . . . . 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 28490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No) → ((𝐴 +s ( 1s /su 𝑛)) −s 𝐴) = ( 1s /su 𝑛))
107	51, 58, 106	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → ((𝐴 +s ( 1s /su 𝑛)) −s 𝐴) = ( 1s /su 𝑛))
108		pncan2s 28490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 ∈ No ∧ ( 1s /su 𝑚) ∈ No) → ((𝐵 +s ( 1s /su 𝑚)) −s 𝐵) = ( 1s /su 𝑚))
109	60, 65, 108	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑛 ∈ ℕs) ∧ 𝑚 ∈ ℕs) → ((𝐵 +s ( 1s /su 𝑚)) −s 𝐵) = ( 1s /su 𝑚))
110	107, 109	oveq12d 7461	. . . . . . . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . 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 286	. . . . . . . . . . . . . . . 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 3188	. . . . . . . . . . . . . . 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 286	. . . . . . . . . . . . . 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 3188	. . . . . . . . . . . . 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 290	. . . . . . . . . . . 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 286	. . . . . . . . . . 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 2829	. . . . . . . . . 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 4171	. . . . . . . . 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 4159	. . . . . . . . 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 2814	. . . . . . . 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 3196	. . . . . . . . . . . . . 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 1881	. . . . . . . . . . . . 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 3293	. . . . . . . . . . . . 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 3656	. . . . . . . . . . . . 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 307	. . . . . . . . . . . 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 7458	. . . . . . . . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . 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 3190	. . . . . . . . . . . . . . . . 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 3502	. . . . . . . . . . . . . . . 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 3196	. . . . . . . . . . . . . . . . . 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 1881	. . . . . . . . . . . . . . . . 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 3293	. . . . . . . . . . . . . . . . 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 3656	. . . . . . . . . . . . . . . . 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 307	. . . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . 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 3502	. . . . . . . . . . . . . . . . 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 7461	. . . . . . . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . 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 286	. . . . . . . . . . . . . . . 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 3188	. . . . . . . . . . . . . . 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 286	. . . . . . . . . . . . . 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 3188	. . . . . . . . . . . . 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 28916	. . . . . . . . . . . . 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 290	. . . . . . . . . . . 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 286	. . . . . . . . . . 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 2829	. . . . . . . . . 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 3196	. . . . . . . . . . . . . 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 1881	. . . . . . . . . . . . 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 3293	. . . . . . . . . . . . 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 3656	. . . . . . . . . . . . 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 307	. . . . . . . . . . . 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 7458	. . . . . . . . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . 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 3190	. . . . . . . . . . . . . . . . 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 3502	. . . . . . . . . . . . . . . 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 3196	. . . . . . . . . . . . . . . . . 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 1881	. . . . . . . . . . . . . . . . 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 3293	. . . . . . . . . . . . . . . . 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 3656	. . . . . . . . . . . . . . . . 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 307	. . . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . 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 3502	. . . . . . . . . . . . . . . . 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 7461	. . . . . . . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . 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 286	. . . . . . . . . . . . . . . 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 3188	. . . . . . . . . . . . . . 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 286	. . . . . . . . . . . . . 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 3188	. . . . . . . . . . . . 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 290	. . . . . . . . . . . 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 286	. . . . . . . . . . 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 2829	. . . . . . . . . 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 4171	. . . . . . . . 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 4159	. . . . . . . . 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 2814	. . . . . . . 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 7461	. . . . . . 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 486	. . . . . 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 2798	. . . . 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 605	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ ((∃𝑛 ∈ ℕs (( -s‘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 (( -s‘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 525	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ ((∃𝑛 ∈ ℕs (( -s‘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 (( -s‘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 (( -s‘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 673	. 2 ⊢ (((𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -s‘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 (( -s‘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 (( -s‘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 28907	. . 3 ⊢ (𝐴 ∈ ℝs ↔ (𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -s‘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 28907	. . 3 ⊢ (𝐵 ∈ ℝs ↔ (𝐵 ∈ No ∧ (∃𝑚 ∈ ℕs (( -s‘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 640	. 2 ⊢ ((𝐴 ∈ ℝs ∧ 𝐵 ∈ ℝs) ↔ ((𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -s‘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 (( -s‘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 28907	. 2 ⊢ ((𝐴 ·s 𝐵) ∈ ℝs ↔ ((𝐴 ·s 𝐵) ∈ No ∧ (∃𝑝 ∈ ℕs (( -s‘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 295	1 ⊢ ((𝐴 ∈ ℝs ∧ 𝐵 ∈ ℝs) → (𝐴 ·s 𝐵) ∈ ℝs)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∃wrex 3090   ∪ cun 4152   class class class wbr 5106  ‘cfv 6558  (class class class)co 7443  Nocsur 28027   <s clts 28028   <<s cslts 28173   |s ccuts 28175   0_s c0s 28221   1_s c1s 28222   +_s cadds 28375   -_scnegs 28435   −_s csubs 28436   ·_s cmuls 28522   /_su cdivs 28603  ℕ_scnns 28729  ℝ_screno 28905

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669  ax-dc 10551

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-oadd 8508  df-nadd 8703  df-no 28030  df-lts 28031  df-bday 28032  df-les 28132  df-slts 28174  df-cuts 28176  df-0s 28223  df-1s 28224  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec 28354  df-norec2 28365  df-adds 28376  df-negs 28437  df-subs 28438  df-muls 28523  df-divs 28604  df-abss 28654  df-n0s 28730  df-nns 28731  df-reno 28906

This theorem is used by: (None)