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Theorem readdscl 28240
Description: The surreal reals are closed under addition. Part of theorem 13(ii) of [Conway] p. 24. (Contributed by Scott Fenton, 15-Apr-2025.)
Assertion
Ref Expression
readdscl ((𝐴 ∈ ℝs𝐵 ∈ ℝs) → (𝐴 +s 𝐵) ∈ ℝs)

Proof of Theorem readdscl
Dummy variables 𝑛 𝑚 𝑝 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addscl 27911 . . . . 5 ((𝐴 No 𝐵 No ) → (𝐴 +s 𝐵) ∈ No )
21adantr 480 . . . 4 (((𝐴 No 𝐵 No ) ∧ ((∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us𝑚) <s 𝐵𝐵 <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 nnaddscl 28225 . . . . . . . . . 10 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → (𝑛 +s 𝑚) ∈ ℕs)
43adantr 480 . . . . . . . . 9 (((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚))) → (𝑛 +s 𝑚) ∈ ℕs)
54adantl 481 . . . . . . . 8 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → (𝑛 +s 𝑚) ∈ ℕs)
6 simprll 778 . . . . . . . . . . 11 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → 𝑛 ∈ ℕs)
76nnsnod 28211 . . . . . . . . . 10 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → 𝑛 No )
8 simprlr 779 . . . . . . . . . . 11 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → 𝑚 ∈ ℕs)
98nnsnod 28211 . . . . . . . . . 10 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → 𝑚 No )
10 negsdi 27975 . . . . . . . . . 10 ((𝑛 No 𝑚 No ) → ( -us ‘(𝑛 +s 𝑚)) = (( -us𝑛) +s ( -us𝑚)))
117, 9, 10syl2anc 583 . . . . . . . . 9 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → ( -us ‘(𝑛 +s 𝑚)) = (( -us𝑛) +s ( -us𝑚)))
127negscld 27962 . . . . . . . . . 10 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → ( -us𝑛) ∈ No )
139negscld 27962 . . . . . . . . . 10 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → ( -us𝑚) ∈ No )
14 simpll 766 . . . . . . . . . 10 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → 𝐴 No )
15 simplr 768 . . . . . . . . . 10 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → 𝐵 No )
16 simprll 778 . . . . . . . . . . 11 (((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚))) → ( -us𝑛) <s 𝐴)
1716adantl 481 . . . . . . . . . 10 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → ( -us𝑛) <s 𝐴)
18 simprrl 780 . . . . . . . . . . 11 (((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚))) → ( -us𝑚) <s 𝐵)
1918adantl 481 . . . . . . . . . 10 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → ( -us𝑚) <s 𝐵)
2012, 13, 14, 15, 17, 19slt2addd 27943 . . . . . . . . 9 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → (( -us𝑛) +s ( -us𝑚)) <s (𝐴 +s 𝐵))
2111, 20eqbrtrd 5170 . . . . . . . 8 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → ( -us ‘(𝑛 +s 𝑚)) <s (𝐴 +s 𝐵))
22 simprlr 779 . . . . . . . . . 10 (((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚))) → 𝐴 <s 𝑛)
2322adantl 481 . . . . . . . . 9 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → 𝐴 <s 𝑛)
24 simprrr 781 . . . . . . . . . 10 (((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚))) → 𝐵 <s 𝑚)
2524adantl 481 . . . . . . . . 9 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → 𝐵 <s 𝑚)
2614, 15, 7, 9, 23, 25slt2addd 27943 . . . . . . . 8 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → (𝐴 +s 𝐵) <s (𝑛 +s 𝑚))
27 fveq2 6897 . . . . . . . . . . 11 (𝑝 = (𝑛 +s 𝑚) → ( -us𝑝) = ( -us ‘(𝑛 +s 𝑚)))
2827breq1d 5158 . . . . . . . . . 10 (𝑝 = (𝑛 +s 𝑚) → (( -us𝑝) <s (𝐴 +s 𝐵) ↔ ( -us ‘(𝑛 +s 𝑚)) <s (𝐴 +s 𝐵)))
29 breq2 5152 . . . . . . . . . 10 (𝑝 = (𝑛 +s 𝑚) → ((𝐴 +s 𝐵) <s 𝑝 ↔ (𝐴 +s 𝐵) <s (𝑛 +s 𝑚)))
3028, 29anbi12d 631 . . . . . . . . 9 (𝑝 = (𝑛 +s 𝑚) → ((( -us𝑝) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s 𝑝) ↔ (( -us ‘(𝑛 +s 𝑚)) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s (𝑛 +s 𝑚))))
3130rspcev 3609 . . . . . . . 8 (((𝑛 +s 𝑚) ∈ ℕs ∧ (( -us ‘(𝑛 +s 𝑚)) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s (𝑛 +s 𝑚))) → ∃𝑝 ∈ ℕs (( -us𝑝) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s 𝑝))
325, 21, 26, 31syl12anc 836 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ ((𝑛 ∈ ℕs𝑚 ∈ ℕs) ∧ ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))) → ∃𝑝 ∈ ℕs (( -us𝑝) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s 𝑝))
3332expr 456 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ (𝑛 ∈ ℕs𝑚 ∈ ℕs)) → (((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)) → ∃𝑝 ∈ ℕs (( -us𝑝) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s 𝑝)))
3433rexlimdvva 3208 . . . . 5 ((𝐴 No 𝐵 No ) → (∃𝑛 ∈ ℕs𝑚 ∈ ℕs ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)) → ∃𝑝 ∈ ℕs (( -us𝑝) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s 𝑝)))
35 simpl 482 . . . . . . 7 ((∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) → ∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛))
36 simpl 482 . . . . . . 7 ((∃𝑚 ∈ ℕs (( -us𝑚) <s 𝐵𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})) → ∃𝑚 ∈ ℕs (( -us𝑚) <s 𝐵𝐵 <s 𝑚))
3735, 36anim12i 612 . . . . . 6 (((∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us𝑚) <s 𝐵𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → (∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ ∃𝑚 ∈ ℕs (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))
38 reeanv 3223 . . . . . 6 (∃𝑛 ∈ ℕs𝑚 ∈ ℕs ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)) ↔ (∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ ∃𝑚 ∈ ℕs (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))
3937, 38sylibr 233 . . . . 5 (((∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us𝑚) <s 𝐵𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → ∃𝑛 ∈ ℕs𝑚 ∈ ℕs ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ (( -us𝑚) <s 𝐵𝐵 <s 𝑚)))
4034, 39impel 505 . . . 4 (((𝐴 No 𝐵 No ) ∧ ((∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us𝑚) <s 𝐵𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})))) → ∃𝑝 ∈ ℕs (( -us𝑝) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s 𝑝))
41 simpr 484 . . . . . 6 ((∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <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))}))
42 simpr 484 . . . . . 6 ((∃𝑚 ∈ ℕs (( -us𝑚) <s 𝐵𝐵 <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))}))
4341, 42anim12i 612 . . . . 5 (((∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us𝑚) <s 𝐵𝐵 <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))})))
44 simpll 766 . . . . . . . 8 (((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 No )
45 recut 28237 . . . . . . . 8 (𝐴 No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})
4644, 45syl 17 . . . . . . 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))})
47 simplr 768 . . . . . . . 8 (((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 No )
48 recut 28237 . . . . . . . 8 (𝐵 No → {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} <<s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})
4947, 48syl 17 . . . . . . 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))})
50 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))}))
51 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))}))
5246, 49, 50, 51addsunif 27932 . . . . . 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))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)}) |s ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)})))
53 ovex 7453 . . . . . . . . . . . . . . 15 (𝐴 -s ( 1s /su 𝑛)) ∈ V
54 oveq1 7427 . . . . . . . . . . . . . . . 16 (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → (𝑡 +s 𝐵) = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵))
5554eqeq2d 2739 . . . . . . . . . . . . . . 15 (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → (𝑧 = (𝑡 +s 𝐵) ↔ 𝑧 = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵)))
5653, 55ceqsexv 3523 . . . . . . . . . . . . . 14 (∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ 𝑧 = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵))
57 simpll 766 . . . . . . . . . . . . . . . 16 (((𝐴 No 𝐵 No ) ∧ 𝑛 ∈ ℕs) → 𝐴 No )
58 simplr 768 . . . . . . . . . . . . . . . 16 (((𝐴 No 𝐵 No ) ∧ 𝑛 ∈ ℕs) → 𝐵 No )
59 1sno 27773 . . . . . . . . . . . . . . . . . . 19 1s No
6059a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕs → 1s No )
61 nnsno 28209 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕs𝑛 No )
62 nnne0s 28218 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕs𝑛 ≠ 0s )
6360, 61, 62divscld 28135 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
6463adantl 481 . . . . . . . . . . . . . . . 16 (((𝐴 No 𝐵 No ) ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
6557, 58, 64addsubsd 28003 . . . . . . . . . . . . . . 15 (((𝐴 No 𝐵 No ) ∧ 𝑛 ∈ ℕs) → ((𝐴 +s 𝐵) -s ( 1s /su 𝑛)) = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵))
6665eqeq2d 2739 . . . . . . . . . . . . . 14 (((𝐴 No 𝐵 No ) ∧ 𝑛 ∈ ℕs) → (𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑛)) ↔ 𝑧 = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵)))
6756, 66bitr4id 290 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No ) ∧ 𝑛 ∈ ℕs) → (∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑛))))
6867rexbidva 3173 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No ) → (∃𝑛 ∈ ℕs𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ ∃𝑛 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑛))))
69 r19.41v 3185 . . . . . . . . . . . . . 14 (∃𝑛 ∈ ℕs (𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ (∃𝑛 ∈ ℕs 𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
7069exbii 1843 . . . . . . . . . . . . 13 (∃𝑡𝑛 ∈ ℕs (𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ ∃𝑡(∃𝑛 ∈ ℕs 𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
71 rexcom4 3282 . . . . . . . . . . . . 13 (∃𝑛 ∈ ℕs𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ ∃𝑡𝑛 ∈ ℕs (𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
72 eqeq1 2732 . . . . . . . . . . . . . . 15 (𝑥 = 𝑡 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑡 = (𝐴 -s ( 1s /su 𝑛))))
7372rexbidv 3175 . . . . . . . . . . . . . 14 (𝑥 = 𝑡 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑡 = (𝐴 -s ( 1s /su 𝑛))))
7473rexab 3689 . . . . . . . . . . . . 13 (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵) ↔ ∃𝑡(∃𝑛 ∈ ℕs 𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
7570, 71, 743bitr4ri 304 . . . . . . . . . . . 12 (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵) ↔ ∃𝑛 ∈ ℕs𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
76 oveq2 7428 . . . . . . . . . . . . . . 15 (𝑝 = 𝑛 → ( 1s /su 𝑝) = ( 1s /su 𝑛))
7776oveq2d 7436 . . . . . . . . . . . . . 14 (𝑝 = 𝑛 → ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) = ((𝐴 +s 𝐵) -s ( 1s /su 𝑛)))
7877eqeq2d 2739 . . . . . . . . . . . . 13 (𝑝 = 𝑛 → (𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) ↔ 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑛))))
7978cbvrexvw 3232 . . . . . . . . . . . 12 (∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) ↔ ∃𝑛 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑛)))
8068, 75, 793bitr4g 314 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))))
8180abbidv 2797 . . . . . . . . . 10 ((𝐴 No 𝐵 No ) → {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))})
82 ovex 7453 . . . . . . . . . . . . . . 15 (𝐵 -s ( 1s /su 𝑚)) ∈ V
83 oveq2 7428 . . . . . . . . . . . . . . . 16 (𝑡 = (𝐵 -s ( 1s /su 𝑚)) → (𝐴 +s 𝑡) = (𝐴 +s (𝐵 -s ( 1s /su 𝑚))))
8483eqeq2d 2739 . . . . . . . . . . . . . . 15 (𝑡 = (𝐵 -s ( 1s /su 𝑚)) → (𝑧 = (𝐴 +s 𝑡) ↔ 𝑧 = (𝐴 +s (𝐵 -s ( 1s /su 𝑚)))))
8582, 84ceqsexv 3523 . . . . . . . . . . . . . 14 (∃𝑡(𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ 𝑧 = (𝐴 +s (𝐵 -s ( 1s /su 𝑚))))
86 simpll 766 . . . . . . . . . . . . . . . 16 (((𝐴 No 𝐵 No ) ∧ 𝑚 ∈ ℕs) → 𝐴 No )
87 simplr 768 . . . . . . . . . . . . . . . 16 (((𝐴 No 𝐵 No ) ∧ 𝑚 ∈ ℕs) → 𝐵 No )
8859a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕs → 1s No )
89 nnsno 28209 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕs𝑚 No )
90 nnne0s 28218 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕs𝑚 ≠ 0s )
9188, 89, 90divscld 28135 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕs → ( 1s /su 𝑚) ∈ No )
9291adantl 481 . . . . . . . . . . . . . . . 16 (((𝐴 No 𝐵 No ) ∧ 𝑚 ∈ ℕs) → ( 1s /su 𝑚) ∈ No )
9386, 87, 92addsubsassd 28002 . . . . . . . . . . . . . . 15 (((𝐴 No 𝐵 No ) ∧ 𝑚 ∈ ℕs) → ((𝐴 +s 𝐵) -s ( 1s /su 𝑚)) = (𝐴 +s (𝐵 -s ( 1s /su 𝑚))))
9493eqeq2d 2739 . . . . . . . . . . . . . 14 (((𝐴 No 𝐵 No ) ∧ 𝑚 ∈ ℕs) → (𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑚)) ↔ 𝑧 = (𝐴 +s (𝐵 -s ( 1s /su 𝑚)))))
9585, 94bitr4id 290 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No ) ∧ 𝑚 ∈ ℕs) → (∃𝑡(𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑚))))
9695rexbidva 3173 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No ) → (∃𝑚 ∈ ℕs𝑡(𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑚))))
97 r19.41v 3185 . . . . . . . . . . . . . 14 (∃𝑚 ∈ ℕs (𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ (∃𝑚 ∈ ℕs 𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
9897exbii 1843 . . . . . . . . . . . . 13 (∃𝑡𝑚 ∈ ℕs (𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ ∃𝑡(∃𝑚 ∈ ℕs 𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
99 rexcom4 3282 . . . . . . . . . . . . 13 (∃𝑚 ∈ ℕs𝑡(𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ ∃𝑡𝑚 ∈ ℕs (𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
100 eqeq1 2732 . . . . . . . . . . . . . . 15 (𝑦 = 𝑡 → (𝑦 = (𝐵 -s ( 1s /su 𝑚)) ↔ 𝑡 = (𝐵 -s ( 1s /su 𝑚))))
101100rexbidv 3175 . . . . . . . . . . . . . 14 (𝑦 = 𝑡 → (∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚)) ↔ ∃𝑚 ∈ ℕs 𝑡 = (𝐵 -s ( 1s /su 𝑚))))
102101rexab 3689 . . . . . . . . . . . . 13 (∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡) ↔ ∃𝑡(∃𝑚 ∈ ℕs 𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
10398, 99, 1023bitr4ri 304 . . . . . . . . . . . 12 (∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡) ↔ ∃𝑚 ∈ ℕs𝑡(𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
104 oveq2 7428 . . . . . . . . . . . . . . 15 (𝑝 = 𝑚 → ( 1s /su 𝑝) = ( 1s /su 𝑚))
105104oveq2d 7436 . . . . . . . . . . . . . 14 (𝑝 = 𝑚 → ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) = ((𝐴 +s 𝐵) -s ( 1s /su 𝑚)))
106105eqeq2d 2739 . . . . . . . . . . . . 13 (𝑝 = 𝑚 → (𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) ↔ 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑚))))
107106cbvrexvw 3232 . . . . . . . . . . . 12 (∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) ↔ ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑚)))
10896, 103, 1073bitr4g 314 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))))
109108abbidv 2797 . . . . . . . . . 10 ((𝐴 No 𝐵 No ) → {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)} = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))})
11081, 109uneq12d 4163 . . . . . . . . 9 ((𝐴 No 𝐵 No ) → ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)}) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))} ∪ {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))}))
111 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))}
112110, 111eqtrdi 2784 . . . . . . . 8 ((𝐴 No 𝐵 No ) → ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)}) = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))})
113 ovex 7453 . . . . . . . . . . . . . . 15 (𝐴 +s ( 1s /su 𝑛)) ∈ V
114 oveq1 7427 . . . . . . . . . . . . . . . 16 (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → (𝑡 +s 𝐵) = ((𝐴 +s ( 1s /su 𝑛)) +s 𝐵))
115114eqeq2d 2739 . . . . . . . . . . . . . . 15 (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → (𝑧 = (𝑡 +s 𝐵) ↔ 𝑧 = ((𝐴 +s ( 1s /su 𝑛)) +s 𝐵)))
116113, 115ceqsexv 3523 . . . . . . . . . . . . . 14 (∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ 𝑧 = ((𝐴 +s ( 1s /su 𝑛)) +s 𝐵))
11757, 64, 58adds32d 27937 . . . . . . . . . . . . . . 15 (((𝐴 No 𝐵 No ) ∧ 𝑛 ∈ ℕs) → ((𝐴 +s ( 1s /su 𝑛)) +s 𝐵) = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛)))
118117eqeq2d 2739 . . . . . . . . . . . . . 14 (((𝐴 No 𝐵 No ) ∧ 𝑛 ∈ ℕs) → (𝑧 = ((𝐴 +s ( 1s /su 𝑛)) +s 𝐵) ↔ 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛))))
119116, 118bitrid 283 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No ) ∧ 𝑛 ∈ ℕs) → (∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛))))
120119rexbidva 3173 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No ) → (∃𝑛 ∈ ℕs𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ ∃𝑛 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛))))
121 r19.41v 3185 . . . . . . . . . . . . . 14 (∃𝑛 ∈ ℕs (𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ (∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
122121exbii 1843 . . . . . . . . . . . . 13 (∃𝑡𝑛 ∈ ℕs (𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ ∃𝑡(∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
123 rexcom4 3282 . . . . . . . . . . . . 13 (∃𝑛 ∈ ℕs𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ ∃𝑡𝑛 ∈ ℕs (𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
124 eqeq1 2732 . . . . . . . . . . . . . . 15 (𝑥 = 𝑡 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑡 = (𝐴 +s ( 1s /su 𝑛))))
125124rexbidv 3175 . . . . . . . . . . . . . 14 (𝑥 = 𝑡 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛))))
126125rexab 3689 . . . . . . . . . . . . 13 (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵) ↔ ∃𝑡(∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
127122, 123, 1263bitr4ri 304 . . . . . . . . . . . 12 (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵) ↔ ∃𝑛 ∈ ℕs𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
12876oveq2d 7436 . . . . . . . . . . . . . 14 (𝑝 = 𝑛 → ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛)))
129128eqeq2d 2739 . . . . . . . . . . . . 13 (𝑝 = 𝑛 → (𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) ↔ 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛))))
130129cbvrexvw 3232 . . . . . . . . . . . 12 (∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) ↔ ∃𝑛 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛)))
131120, 127, 1303bitr4g 314 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))))
132131abbidv 2797 . . . . . . . . . 10 ((𝐴 No 𝐵 No ) → {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))})
133 ovex 7453 . . . . . . . . . . . . . . 15 (𝐵 +s ( 1s /su 𝑚)) ∈ V
134 oveq2 7428 . . . . . . . . . . . . . . . 16 (𝑡 = (𝐵 +s ( 1s /su 𝑚)) → (𝐴 +s 𝑡) = (𝐴 +s (𝐵 +s ( 1s /su 𝑚))))
135134eqeq2d 2739 . . . . . . . . . . . . . . 15 (𝑡 = (𝐵 +s ( 1s /su 𝑚)) → (𝑧 = (𝐴 +s 𝑡) ↔ 𝑧 = (𝐴 +s (𝐵 +s ( 1s /su 𝑚)))))
136133, 135ceqsexv 3523 . . . . . . . . . . . . . 14 (∃𝑡(𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ 𝑧 = (𝐴 +s (𝐵 +s ( 1s /su 𝑚))))
13786, 87, 92addsassd 27936 . . . . . . . . . . . . . . 15 (((𝐴 No 𝐵 No ) ∧ 𝑚 ∈ ℕs) → ((𝐴 +s 𝐵) +s ( 1s /su 𝑚)) = (𝐴 +s (𝐵 +s ( 1s /su 𝑚))))
138137eqeq2d 2739 . . . . . . . . . . . . . 14 (((𝐴 No 𝐵 No ) ∧ 𝑚 ∈ ℕs) → (𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑚)) ↔ 𝑧 = (𝐴 +s (𝐵 +s ( 1s /su 𝑚)))))
139136, 138bitr4id 290 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No ) ∧ 𝑚 ∈ ℕs) → (∃𝑡(𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑚))))
140139rexbidva 3173 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No ) → (∃𝑚 ∈ ℕs𝑡(𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑚))))
141 r19.41v 3185 . . . . . . . . . . . . . 14 (∃𝑚 ∈ ℕs (𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ (∃𝑚 ∈ ℕs 𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
142141exbii 1843 . . . . . . . . . . . . 13 (∃𝑡𝑚 ∈ ℕs (𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ ∃𝑡(∃𝑚 ∈ ℕs 𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
143 rexcom4 3282 . . . . . . . . . . . . 13 (∃𝑚 ∈ ℕs𝑡(𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ ∃𝑡𝑚 ∈ ℕs (𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
144 eqeq1 2732 . . . . . . . . . . . . . . 15 (𝑦 = 𝑡 → (𝑦 = (𝐵 +s ( 1s /su 𝑚)) ↔ 𝑡 = (𝐵 +s ( 1s /su 𝑚))))
145144rexbidv 3175 . . . . . . . . . . . . . 14 (𝑦 = 𝑡 → (∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚)) ↔ ∃𝑚 ∈ ℕs 𝑡 = (𝐵 +s ( 1s /su 𝑚))))
146145rexab 3689 . . . . . . . . . . . . 13 (∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡) ↔ ∃𝑡(∃𝑚 ∈ ℕs 𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
147142, 143, 1463bitr4ri 304 . . . . . . . . . . . 12 (∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡) ↔ ∃𝑚 ∈ ℕs𝑡(𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
148104oveq2d 7436 . . . . . . . . . . . . . 14 (𝑝 = 𝑚 → ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) = ((𝐴 +s 𝐵) +s ( 1s /su 𝑚)))
149148eqeq2d 2739 . . . . . . . . . . . . 13 (𝑝 = 𝑚 → (𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) ↔ 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑚))))
150149cbvrexvw 3232 . . . . . . . . . . . 12 (∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) ↔ ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑚)))
151140, 147, 1503bitr4g 314 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))))
152151abbidv 2797 . . . . . . . . . 10 ((𝐴 No 𝐵 No ) → {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)} = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))})
153132, 152uneq12d 4163 . . . . . . . . 9 ((𝐴 No 𝐵 No ) → ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)}) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))} ∪ {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))}))
154 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))}
155153, 154eqtrdi 2784 . . . . . . . 8 ((𝐴 No 𝐵 No ) → ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)}) = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))})
156112, 155oveq12d 7438 . . . . . . 7 ((𝐴 No 𝐵 No ) → (({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)}) |s ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)})) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))} |s {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))}))
157156adantr 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))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)}) |s ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)})) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))} |s {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))}))
15852, 157eqtrd 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))}))
15943, 158sylan2 592 . . . 4 (((𝐴 No 𝐵 No ) ∧ ((∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us𝑚) <s 𝐵𝐵 <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))}))
1602, 40, 159jca32 515 . . 3 (((𝐴 No 𝐵 No ) ∧ ((∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us𝑚) <s 𝐵𝐵 <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 (( -us𝑝) <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))}))))
161160an4s 659 . 2 (((𝐴 No ∧ (∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <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 (( -us𝑚) <s 𝐵𝐵 <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 (( -us𝑝) <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))}))))
162 elreno 28236 . . 3 (𝐴 ∈ ℝs ↔ (𝐴 No ∧ (∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))))
163 elreno 28236 . . 3 (𝐵 ∈ ℝs ↔ (𝐵 No ∧ (∃𝑚 ∈ ℕs (( -us𝑚) <s 𝐵𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))))
164162, 163anbi12i 627 . 2 ((𝐴 ∈ ℝs𝐵 ∈ ℝs) ↔ ((𝐴 No ∧ (∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <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 (( -us𝑚) <s 𝐵𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})))))
165 elreno 28236 . 2 ((𝐴 +s 𝐵) ∈ ℝs ↔ ((𝐴 +s 𝐵) ∈ No ∧ (∃𝑝 ∈ ℕs (( -us𝑝) <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))}))))
166161, 164, 1653imtr4i 292 1 ((𝐴 ∈ ℝs𝐵 ∈ ℝs) → (𝐴 +s 𝐵) ∈ ℝs)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wex 1774  wcel 2099  {cab 2705  wrex 3067  cun 3945   class class class wbr 5148  cfv 6548  (class class class)co 7420   No csur 27586   <s cslt 27587   <<s csslt 27726   |s cscut 27728   1s c1s 27769   +s cadds 27889   -us cnegs 27945   -s csubs 27946   /su cdivs 28100  scnns 28199  screno 28234
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 10470
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 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-oadd 8491  df-nadd 8687  df-no 27589  df-slt 27590  df-bday 27591  df-sle 27691  df-sslt 27727  df-scut 27729  df-0s 27770  df-1s 27771  df-made 27787  df-old 27788  df-left 27790  df-right 27791  df-norec 27868  df-norec2 27879  df-adds 27890  df-negs 27947  df-subs 27948  df-muls 28020  df-divs 28101  df-n0s 28200  df-nns 28201  df-reno 28235
This theorem is referenced by: (None)
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