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Theorem addsasslem2 27315
Description: Lemma for addition associativity. Expand the other form of the triple sum. (Contributed by Scott Fenton, 21-Jan-2025.)
Hypotheses
Ref Expression
addsasslem.1 (𝜑𝐴 No )
addsasslem.2 (𝜑𝐵 No )
addsasslem.3 (𝜑𝐶 No )
Assertion
Ref Expression
addsasslem2 (𝜑 → (𝐴 +s (𝐵 +s 𝐶)) = ((({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))}) |s (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))})))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑐,𝑙,𝑚,𝑛,𝑝,𝑞,𝑟,𝑤,𝑦,𝑧   𝐵,𝑎,𝑏,𝑐,𝑙,𝑚,𝑛,𝑝,𝑞,𝑟,𝑤,𝑦,𝑧   𝐶,𝑎,𝑏,𝑐,𝑙,𝑚,𝑛,𝑝,𝑞,𝑟,𝑤,𝑦,𝑧   𝜑,𝑎,𝑏,𝑐,𝑙,𝑚,𝑛,𝑝,𝑞,𝑟,𝑤,𝑦,𝑧

Proof of Theorem addsasslem2
Dummy variables 𝑑 𝑒 𝑓 𝑔 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addsasslem.1 . . . 4 (𝜑𝐴 No )
2 lltropt 27205 . . . 4 (𝐴 No → ( L ‘𝐴) <<s ( R ‘𝐴))
31, 2syl 17 . . 3 (𝜑 → ( L ‘𝐴) <<s ( R ‘𝐴))
4 addsasslem.2 . . . . . 6 (𝜑𝐵 No )
5 addsasslem.3 . . . . . 6 (𝜑𝐶 No )
64, 5addscut 27292 . . . . 5 (𝜑 → ((𝐵 +s 𝐶) ∈ No ∧ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) <<s {(𝐵 +s 𝐶)} ∧ {(𝐵 +s 𝐶)} <<s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})))
76simp2d 1144 . . . 4 (𝜑 → ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) <<s {(𝐵 +s 𝐶)})
86simp3d 1145 . . . 4 (𝜑 → {(𝐵 +s 𝐶)} <<s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}))
9 ovex 7391 . . . . . 6 (𝐵 +s 𝐶) ∈ V
109snnz 4738 . . . . 5 {(𝐵 +s 𝐶)} ≠ ∅
11 sslttr 27149 . . . . 5 ((({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) <<s {(𝐵 +s 𝐶)} ∧ {(𝐵 +s 𝐶)} <<s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}) ∧ {(𝐵 +s 𝐶)} ≠ ∅) → ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) <<s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}))
1210, 11mp3an3 1451 . . . 4 ((({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) <<s {(𝐵 +s 𝐶)} ∧ {(𝐵 +s 𝐶)} <<s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})) → ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) <<s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}))
137, 8, 12syl2anc 585 . . 3 (𝜑 → ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) <<s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}))
14 lrcut 27235 . . . . 5 (𝐴 No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
151, 14syl 17 . . . 4 (𝜑 → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
1615eqcomd 2743 . . 3 (𝜑𝐴 = (( L ‘𝐴) |s ( R ‘𝐴)))
17 addsval2 27278 . . . 4 ((𝐵 No 𝐶 No ) → (𝐵 +s 𝐶) = (({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) |s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})))
184, 5, 17syl2anc 585 . . 3 (𝜑 → (𝐵 +s 𝐶) = (({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) |s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})))
193, 13, 16, 18addsunif 27313 . 2 (𝜑 → (𝐴 +s (𝐵 +s 𝐶)) = (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃ ∈ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)})𝑧 = (𝐴 +s )}) |s ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})𝑏 = (𝐴 +s 𝑖)})))
20 rexun 4151 . . . . . . . 8 (∃ ∈ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)})𝑧 = (𝐴 +s ) ↔ (∃ ∈ {𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)}𝑧 = (𝐴 +s ) ∨ ∃ ∈ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}𝑧 = (𝐴 +s )))
21 eqeq1 2741 . . . . . . . . . . . 12 (𝑑 = → (𝑑 = (𝑚 +s 𝐶) ↔ = (𝑚 +s 𝐶)))
2221rexbidv 3176 . . . . . . . . . . 11 (𝑑 = → (∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶) ↔ ∃𝑚 ∈ ( L ‘𝐵) = (𝑚 +s 𝐶)))
2322rexab 3653 . . . . . . . . . 10 (∃ ∈ {𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)}𝑧 = (𝐴 +s ) ↔ ∃(∃𝑚 ∈ ( L ‘𝐵) = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s )))
24 rexcom4 3272 . . . . . . . . . . 11 (∃𝑚 ∈ ( L ‘𝐵)∃( = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s )) ↔ ∃𝑚 ∈ ( L ‘𝐵)( = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s )))
25 ovex 7391 . . . . . . . . . . . . 13 (𝑚 +s 𝐶) ∈ V
26 oveq2 7366 . . . . . . . . . . . . . 14 ( = (𝑚 +s 𝐶) → (𝐴 +s ) = (𝐴 +s (𝑚 +s 𝐶)))
2726eqeq2d 2748 . . . . . . . . . . . . 13 ( = (𝑚 +s 𝐶) → (𝑧 = (𝐴 +s ) ↔ 𝑧 = (𝐴 +s (𝑚 +s 𝐶))))
2825, 27ceqsexv 3495 . . . . . . . . . . . 12 (∃( = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s )) ↔ 𝑧 = (𝐴 +s (𝑚 +s 𝐶)))
2928rexbii 3098 . . . . . . . . . . 11 (∃𝑚 ∈ ( L ‘𝐵)∃( = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s )) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶)))
30 r19.41v 3186 . . . . . . . . . . . 12 (∃𝑚 ∈ ( L ‘𝐵)( = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s )) ↔ (∃𝑚 ∈ ( L ‘𝐵) = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s )))
3130exbii 1851 . . . . . . . . . . 11 (∃𝑚 ∈ ( L ‘𝐵)( = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s )) ↔ ∃(∃𝑚 ∈ ( L ‘𝐵) = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s )))
3224, 29, 313bitr3ri 302 . . . . . . . . . 10 (∃(∃𝑚 ∈ ( L ‘𝐵) = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s )) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶)))
3323, 32bitri 275 . . . . . . . . 9 (∃ ∈ {𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)}𝑧 = (𝐴 +s ) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶)))
34 eqeq1 2741 . . . . . . . . . . . 12 (𝑒 = → (𝑒 = (𝐵 +s 𝑛) ↔ = (𝐵 +s 𝑛)))
3534rexbidv 3176 . . . . . . . . . . 11 (𝑒 = → (∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛) ↔ ∃𝑛 ∈ ( L ‘𝐶) = (𝐵 +s 𝑛)))
3635rexab 3653 . . . . . . . . . 10 (∃ ∈ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}𝑧 = (𝐴 +s ) ↔ ∃(∃𝑛 ∈ ( L ‘𝐶) = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s )))
37 rexcom4 3272 . . . . . . . . . . 11 (∃𝑛 ∈ ( L ‘𝐶)∃( = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s )) ↔ ∃𝑛 ∈ ( L ‘𝐶)( = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s )))
38 ovex 7391 . . . . . . . . . . . . 13 (𝐵 +s 𝑛) ∈ V
39 oveq2 7366 . . . . . . . . . . . . . 14 ( = (𝐵 +s 𝑛) → (𝐴 +s ) = (𝐴 +s (𝐵 +s 𝑛)))
4039eqeq2d 2748 . . . . . . . . . . . . 13 ( = (𝐵 +s 𝑛) → (𝑧 = (𝐴 +s ) ↔ 𝑧 = (𝐴 +s (𝐵 +s 𝑛))))
4138, 40ceqsexv 3495 . . . . . . . . . . . 12 (∃( = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s )) ↔ 𝑧 = (𝐴 +s (𝐵 +s 𝑛)))
4241rexbii 3098 . . . . . . . . . . 11 (∃𝑛 ∈ ( L ‘𝐶)∃( = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s )) ↔ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛)))
43 r19.41v 3186 . . . . . . . . . . . 12 (∃𝑛 ∈ ( L ‘𝐶)( = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s )) ↔ (∃𝑛 ∈ ( L ‘𝐶) = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s )))
4443exbii 1851 . . . . . . . . . . 11 (∃𝑛 ∈ ( L ‘𝐶)( = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s )) ↔ ∃(∃𝑛 ∈ ( L ‘𝐶) = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s )))
4537, 42, 443bitr3ri 302 . . . . . . . . . 10 (∃(∃𝑛 ∈ ( L ‘𝐶) = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s )) ↔ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛)))
4636, 45bitri 275 . . . . . . . . 9 (∃ ∈ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}𝑧 = (𝐴 +s ) ↔ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛)))
4733, 46orbi12i 914 . . . . . . . 8 ((∃ ∈ {𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)}𝑧 = (𝐴 +s ) ∨ ∃ ∈ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}𝑧 = (𝐴 +s )) ↔ (∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶)) ∨ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛))))
4820, 47bitri 275 . . . . . . 7 (∃ ∈ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)})𝑧 = (𝐴 +s ) ↔ (∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶)) ∨ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛))))
4948abbii 2807 . . . . . 6 {𝑧 ∣ ∃ ∈ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)})𝑧 = (𝐴 +s )} = {𝑧 ∣ (∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶)) ∨ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛)))}
50 unab 4259 . . . . . 6 ({𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))} ∪ {𝑧 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛))}) = {𝑧 ∣ (∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶)) ∨ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛)))}
51 eqeq1 2741 . . . . . . . . 9 (𝑧 = 𝑤 → (𝑧 = (𝐴 +s (𝐵 +s 𝑛)) ↔ 𝑤 = (𝐴 +s (𝐵 +s 𝑛))))
5251rexbidv 3176 . . . . . . . 8 (𝑧 = 𝑤 → (∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛)) ↔ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))))
5352cbvabv 2810 . . . . . . 7 {𝑧 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛))} = {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))}
5453uneq2i 4121 . . . . . 6 ({𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))} ∪ {𝑧 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛))}) = ({𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))} ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))})
5549, 50, 543eqtr2i 2771 . . . . 5 {𝑧 ∣ ∃ ∈ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)})𝑧 = (𝐴 +s )} = ({𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))} ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))})
5655uneq2i 4121 . . . 4 ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃ ∈ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)})𝑧 = (𝐴 +s )}) = ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ ({𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))} ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))}))
57 unass 4127 . . . 4 (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))}) = ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ ({𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))} ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))}))
5856, 57eqtr4i 2768 . . 3 ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃ ∈ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)})𝑧 = (𝐴 +s )}) = (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))})
59 rexun 4151 . . . . . . . 8 (∃𝑖 ∈ ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})𝑏 = (𝐴 +s 𝑖) ↔ (∃𝑖 ∈ {𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)}𝑏 = (𝐴 +s 𝑖) ∨ ∃𝑖 ∈ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}𝑏 = (𝐴 +s 𝑖)))
60 eqeq1 2741 . . . . . . . . . . . 12 (𝑓 = 𝑖 → (𝑓 = (𝑞 +s 𝐶) ↔ 𝑖 = (𝑞 +s 𝐶)))
6160rexbidv 3176 . . . . . . . . . . 11 (𝑓 = 𝑖 → (∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝑞 +s 𝐶)))
6261rexab 3653 . . . . . . . . . 10 (∃𝑖 ∈ {𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)}𝑏 = (𝐴 +s 𝑖) ↔ ∃𝑖(∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)))
63 rexcom4 3272 . . . . . . . . . . 11 (∃𝑞 ∈ ( R ‘𝐵)∃𝑖(𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ ∃𝑖𝑞 ∈ ( R ‘𝐵)(𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)))
64 ovex 7391 . . . . . . . . . . . . 13 (𝑞 +s 𝐶) ∈ V
65 oveq2 7366 . . . . . . . . . . . . . 14 (𝑖 = (𝑞 +s 𝐶) → (𝐴 +s 𝑖) = (𝐴 +s (𝑞 +s 𝐶)))
6665eqeq2d 2748 . . . . . . . . . . . . 13 (𝑖 = (𝑞 +s 𝐶) → (𝑏 = (𝐴 +s 𝑖) ↔ 𝑏 = (𝐴 +s (𝑞 +s 𝐶))))
6764, 66ceqsexv 3495 . . . . . . . . . . . 12 (∃𝑖(𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ 𝑏 = (𝐴 +s (𝑞 +s 𝐶)))
6867rexbii 3098 . . . . . . . . . . 11 (∃𝑞 ∈ ( R ‘𝐵)∃𝑖(𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶)))
69 r19.41v 3186 . . . . . . . . . . . 12 (∃𝑞 ∈ ( R ‘𝐵)(𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ (∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)))
7069exbii 1851 . . . . . . . . . . 11 (∃𝑖𝑞 ∈ ( R ‘𝐵)(𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ ∃𝑖(∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)))
7163, 68, 703bitr3ri 302 . . . . . . . . . 10 (∃𝑖(∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶)))
7262, 71bitri 275 . . . . . . . . 9 (∃𝑖 ∈ {𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)}𝑏 = (𝐴 +s 𝑖) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶)))
73 eqeq1 2741 . . . . . . . . . . . 12 (𝑔 = 𝑖 → (𝑔 = (𝐵 +s 𝑟) ↔ 𝑖 = (𝐵 +s 𝑟)))
7473rexbidv 3176 . . . . . . . . . . 11 (𝑔 = 𝑖 → (∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟) ↔ ∃𝑟 ∈ ( R ‘𝐶)𝑖 = (𝐵 +s 𝑟)))
7574rexab 3653 . . . . . . . . . 10 (∃𝑖 ∈ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}𝑏 = (𝐴 +s 𝑖) ↔ ∃𝑖(∃𝑟 ∈ ( R ‘𝐶)𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)))
76 rexcom4 3272 . . . . . . . . . . 11 (∃𝑟 ∈ ( R ‘𝐶)∃𝑖(𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ ∃𝑖𝑟 ∈ ( R ‘𝐶)(𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)))
77 ovex 7391 . . . . . . . . . . . . 13 (𝐵 +s 𝑟) ∈ V
78 oveq2 7366 . . . . . . . . . . . . . 14 (𝑖 = (𝐵 +s 𝑟) → (𝐴 +s 𝑖) = (𝐴 +s (𝐵 +s 𝑟)))
7978eqeq2d 2748 . . . . . . . . . . . . 13 (𝑖 = (𝐵 +s 𝑟) → (𝑏 = (𝐴 +s 𝑖) ↔ 𝑏 = (𝐴 +s (𝐵 +s 𝑟))))
8077, 79ceqsexv 3495 . . . . . . . . . . . 12 (∃𝑖(𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ 𝑏 = (𝐴 +s (𝐵 +s 𝑟)))
8180rexbii 3098 . . . . . . . . . . 11 (∃𝑟 ∈ ( R ‘𝐶)∃𝑖(𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟)))
82 r19.41v 3186 . . . . . . . . . . . 12 (∃𝑟 ∈ ( R ‘𝐶)(𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ (∃𝑟 ∈ ( R ‘𝐶)𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)))
8382exbii 1851 . . . . . . . . . . 11 (∃𝑖𝑟 ∈ ( R ‘𝐶)(𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ ∃𝑖(∃𝑟 ∈ ( R ‘𝐶)𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)))
8476, 81, 833bitr3ri 302 . . . . . . . . . 10 (∃𝑖(∃𝑟 ∈ ( R ‘𝐶)𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟)))
8575, 84bitri 275 . . . . . . . . 9 (∃𝑖 ∈ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}𝑏 = (𝐴 +s 𝑖) ↔ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟)))
8672, 85orbi12i 914 . . . . . . . 8 ((∃𝑖 ∈ {𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)}𝑏 = (𝐴 +s 𝑖) ∨ ∃𝑖 ∈ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}𝑏 = (𝐴 +s 𝑖)) ↔ (∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶)) ∨ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟))))
8759, 86bitri 275 . . . . . . 7 (∃𝑖 ∈ ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})𝑏 = (𝐴 +s 𝑖) ↔ (∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶)) ∨ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟))))
8887abbii 2807 . . . . . 6 {𝑏 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})𝑏 = (𝐴 +s 𝑖)} = {𝑏 ∣ (∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶)) ∨ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟)))}
89 unab 4259 . . . . . 6 ({𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟))}) = {𝑏 ∣ (∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶)) ∨ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟)))}
90 eqeq1 2741 . . . . . . . . 9 (𝑏 = 𝑐 → (𝑏 = (𝐴 +s (𝐵 +s 𝑟)) ↔ 𝑐 = (𝐴 +s (𝐵 +s 𝑟))))
9190rexbidv 3176 . . . . . . . 8 (𝑏 = 𝑐 → (∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟)) ↔ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))))
9291cbvabv 2810 . . . . . . 7 {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟))} = {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))}
9392uneq2i 4121 . . . . . 6 ({𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟))}) = ({𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))} ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))})
9488, 89, 933eqtr2i 2771 . . . . 5 {𝑏 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})𝑏 = (𝐴 +s 𝑖)} = ({𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))} ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))})
9594uneq2i 4121 . . . 4 ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})𝑏 = (𝐴 +s 𝑖)}) = ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ ({𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))} ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))}))
96 unass 4127 . . . 4 (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))}) = ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ ({𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))} ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))}))
9795, 96eqtr4i 2768 . . 3 ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})𝑏 = (𝐴 +s 𝑖)}) = (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))})
9858, 97oveq12i 7370 . 2 (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃ ∈ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)})𝑧 = (𝐴 +s )}) |s ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})𝑏 = (𝐴 +s 𝑖)})) = ((({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))}) |s (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))}))
9919, 98eqtrdi 2793 1 (𝜑 → (𝐴 +s (𝐵 +s 𝐶)) = ((({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))}) |s (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846   = wceq 1542  wex 1782  wcel 2107  {cab 2714  wne 2944  wrex 3074  cun 3909  c0 4283  {csn 4587   class class class wbr 5106  cfv 6497  (class class class)co 7358   No csur 26991   <<s csslt 27123   |s cscut 27125   L cleft 27178   R cright 27179   +s cadds 27274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-1o 8413  df-2o 8414  df-nadd 8613  df-no 26994  df-slt 26995  df-bday 26996  df-sle 27096  df-sslt 27124  df-scut 27126  df-0s 27166  df-made 27180  df-old 27181  df-left 27183  df-right 27184  df-norec2 27264  df-adds 27275
This theorem is referenced by:  addsass  27316
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