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Theorem addsasslem1 28037
Description: Lemma for addition associativity. Expand one 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
addsasslem1 (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 addsasslem1
Dummy variables 𝑑 𝑒 𝑓 𝑔 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addsasslem.1 . . . . . 6 (𝜑𝐴 No )
2 addsasslem.2 . . . . . 6 (𝜑𝐵 No )
31, 2addscut 28012 . . . . 5 (𝜑 → ((𝐴 +s 𝐵) ∈ No ∧ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s {(𝐴 +s 𝐵)} ∧ {(𝐴 +s 𝐵)} <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})))
43simp2d 1143 . . . 4 (𝜑 → ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s {(𝐴 +s 𝐵)})
53simp3d 1144 . . . 4 (𝜑 → {(𝐴 +s 𝐵)} <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}))
6 ovex 7465 . . . . . 6 (𝐴 +s 𝐵) ∈ V
76snnz 4775 . . . . 5 {(𝐴 +s 𝐵)} ≠ ∅
8 sslttr 27853 . . . . 5 ((({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s {(𝐴 +s 𝐵)} ∧ {(𝐴 +s 𝐵)} <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}) ∧ {(𝐴 +s 𝐵)} ≠ ∅) → ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}))
97, 8mp3an3 1451 . . . 4 ((({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s {(𝐴 +s 𝐵)} ∧ {(𝐴 +s 𝐵)} <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})) → ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}))
104, 5, 9syl2anc 584 . . 3 (𝜑 → ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}))
11 lltropt 27912 . . . 4 ( L ‘𝐶) <<s ( R ‘𝐶)
1211a1i 11 . . 3 (𝜑 → ( L ‘𝐶) <<s ( R ‘𝐶))
13 addsval2 27997 . . . 4 ((𝐴 No 𝐵 No ) → (𝐴 +s 𝐵) = (({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) |s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})))
141, 2, 13syl2anc 584 . . 3 (𝜑 → (𝐴 +s 𝐵) = (({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) |s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})))
15 addsasslem.3 . . . . 5 (𝜑𝐶 No )
16 lrcut 27942 . . . . 5 (𝐶 No → (( L ‘𝐶) |s ( R ‘𝐶)) = 𝐶)
1715, 16syl 17 . . . 4 (𝜑 → (( L ‘𝐶) |s ( R ‘𝐶)) = 𝐶)
1817eqcomd 2742 . . 3 (𝜑𝐶 = (( L ‘𝐶) |s ( R ‘𝐶)))
1910, 12, 14, 18addsunif 28036 . 2 (𝜑 → ((𝐴 +s 𝐵) +s 𝐶) = (({𝑦 ∣ ∃ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = ( +s 𝐶)} ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = ((𝐴 +s 𝐵) +s 𝑛)}) |s ({𝑎 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶)} ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = ((𝐴 +s 𝐵) +s 𝑟)})))
20 unab 4307 . . . . 5 ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑦 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)}) = {𝑦 ∣ (∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶) ∨ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶))}
21 eqeq1 2740 . . . . . . . 8 (𝑧 = 𝑦 → (𝑧 = ((𝐴 +s 𝑚) +s 𝐶) ↔ 𝑦 = ((𝐴 +s 𝑚) +s 𝐶)))
2221rexbidv 3178 . . . . . . 7 (𝑧 = 𝑦 → (∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)))
2322cbvabv 2811 . . . . . 6 {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶)} = {𝑦 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)}
2423uneq2i 4164 . . . . 5 ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶)}) = ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑦 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)})
25 rexun 4195 . . . . . . 7 (∃ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = ( +s 𝐶) ↔ (∃ ∈ {𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)}𝑦 = ( +s 𝐶) ∨ ∃ ∈ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}𝑦 = ( +s 𝐶)))
26 eqeq1 2740 . . . . . . . . . . 11 (𝑑 = → (𝑑 = (𝑙 +s 𝐵) ↔ = (𝑙 +s 𝐵)))
2726rexbidv 3178 . . . . . . . . . 10 (𝑑 = → (∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵) ↔ ∃𝑙 ∈ ( L ‘𝐴) = (𝑙 +s 𝐵)))
2827rexab 3699 . . . . . . . . 9 (∃ ∈ {𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)}𝑦 = ( +s 𝐶) ↔ ∃(∃𝑙 ∈ ( L ‘𝐴) = (𝑙 +s 𝐵) ∧ 𝑦 = ( +s 𝐶)))
29 rexcom4 3287 . . . . . . . . . 10 (∃𝑙 ∈ ( L ‘𝐴)∃( = (𝑙 +s 𝐵) ∧ 𝑦 = ( +s 𝐶)) ↔ ∃𝑙 ∈ ( L ‘𝐴)( = (𝑙 +s 𝐵) ∧ 𝑦 = ( +s 𝐶)))
30 ovex 7465 . . . . . . . . . . . 12 (𝑙 +s 𝐵) ∈ V
31 oveq1 7439 . . . . . . . . . . . . 13 ( = (𝑙 +s 𝐵) → ( +s 𝐶) = ((𝑙 +s 𝐵) +s 𝐶))
3231eqeq2d 2747 . . . . . . . . . . . 12 ( = (𝑙 +s 𝐵) → (𝑦 = ( +s 𝐶) ↔ 𝑦 = ((𝑙 +s 𝐵) +s 𝐶)))
3330, 32ceqsexv 3531 . . . . . . . . . . 11 (∃( = (𝑙 +s 𝐵) ∧ 𝑦 = ( +s 𝐶)) ↔ 𝑦 = ((𝑙 +s 𝐵) +s 𝐶))
3433rexbii 3093 . . . . . . . . . 10 (∃𝑙 ∈ ( L ‘𝐴)∃( = (𝑙 +s 𝐵) ∧ 𝑦 = ( +s 𝐶)) ↔ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶))
35 r19.41v 3188 . . . . . . . . . . 11 (∃𝑙 ∈ ( L ‘𝐴)( = (𝑙 +s 𝐵) ∧ 𝑦 = ( +s 𝐶)) ↔ (∃𝑙 ∈ ( L ‘𝐴) = (𝑙 +s 𝐵) ∧ 𝑦 = ( +s 𝐶)))
3635exbii 1847 . . . . . . . . . 10 (∃𝑙 ∈ ( L ‘𝐴)( = (𝑙 +s 𝐵) ∧ 𝑦 = ( +s 𝐶)) ↔ ∃(∃𝑙 ∈ ( L ‘𝐴) = (𝑙 +s 𝐵) ∧ 𝑦 = ( +s 𝐶)))
3729, 34, 363bitr3ri 302 . . . . . . . . 9 (∃(∃𝑙 ∈ ( L ‘𝐴) = (𝑙 +s 𝐵) ∧ 𝑦 = ( +s 𝐶)) ↔ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶))
3828, 37bitri 275 . . . . . . . 8 (∃ ∈ {𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)}𝑦 = ( +s 𝐶) ↔ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶))
39 eqeq1 2740 . . . . . . . . . . 11 (𝑒 = → (𝑒 = (𝐴 +s 𝑚) ↔ = (𝐴 +s 𝑚)))
4039rexbidv 3178 . . . . . . . . . 10 (𝑒 = → (∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚) ↔ ∃𝑚 ∈ ( L ‘𝐵) = (𝐴 +s 𝑚)))
4140rexab 3699 . . . . . . . . 9 (∃ ∈ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}𝑦 = ( +s 𝐶) ↔ ∃(∃𝑚 ∈ ( L ‘𝐵) = (𝐴 +s 𝑚) ∧ 𝑦 = ( +s 𝐶)))
42 rexcom4 3287 . . . . . . . . . 10 (∃𝑚 ∈ ( L ‘𝐵)∃( = (𝐴 +s 𝑚) ∧ 𝑦 = ( +s 𝐶)) ↔ ∃𝑚 ∈ ( L ‘𝐵)( = (𝐴 +s 𝑚) ∧ 𝑦 = ( +s 𝐶)))
43 ovex 7465 . . . . . . . . . . . 12 (𝐴 +s 𝑚) ∈ V
44 oveq1 7439 . . . . . . . . . . . . 13 ( = (𝐴 +s 𝑚) → ( +s 𝐶) = ((𝐴 +s 𝑚) +s 𝐶))
4544eqeq2d 2747 . . . . . . . . . . . 12 ( = (𝐴 +s 𝑚) → (𝑦 = ( +s 𝐶) ↔ 𝑦 = ((𝐴 +s 𝑚) +s 𝐶)))
4643, 45ceqsexv 3531 . . . . . . . . . . 11 (∃( = (𝐴 +s 𝑚) ∧ 𝑦 = ( +s 𝐶)) ↔ 𝑦 = ((𝐴 +s 𝑚) +s 𝐶))
4746rexbii 3093 . . . . . . . . . 10 (∃𝑚 ∈ ( L ‘𝐵)∃( = (𝐴 +s 𝑚) ∧ 𝑦 = ( +s 𝐶)) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶))
48 r19.41v 3188 . . . . . . . . . . 11 (∃𝑚 ∈ ( L ‘𝐵)( = (𝐴 +s 𝑚) ∧ 𝑦 = ( +s 𝐶)) ↔ (∃𝑚 ∈ ( L ‘𝐵) = (𝐴 +s 𝑚) ∧ 𝑦 = ( +s 𝐶)))
4948exbii 1847 . . . . . . . . . 10 (∃𝑚 ∈ ( L ‘𝐵)( = (𝐴 +s 𝑚) ∧ 𝑦 = ( +s 𝐶)) ↔ ∃(∃𝑚 ∈ ( L ‘𝐵) = (𝐴 +s 𝑚) ∧ 𝑦 = ( +s 𝐶)))
5042, 47, 493bitr3ri 302 . . . . . . . . 9 (∃(∃𝑚 ∈ ( L ‘𝐵) = (𝐴 +s 𝑚) ∧ 𝑦 = ( +s 𝐶)) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶))
5141, 50bitri 275 . . . . . . . 8 (∃ ∈ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}𝑦 = ( +s 𝐶) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶))
5238, 51orbi12i 914 . . . . . . 7 ((∃ ∈ {𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)}𝑦 = ( +s 𝐶) ∨ ∃ ∈ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}𝑦 = ( +s 𝐶)) ↔ (∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶) ∨ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)))
5325, 52bitri 275 . . . . . 6 (∃ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = ( +s 𝐶) ↔ (∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶) ∨ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)))
5453abbii 2808 . . . . 5 {𝑦 ∣ ∃ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = ( +s 𝐶)} = {𝑦 ∣ (∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶) ∨ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶))}
5520, 24, 543eqtr4ri 2775 . . . 4 {𝑦 ∣ ∃ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = ( +s 𝐶)} = ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶)})
5655uneq1i 4163 . . 3 ({𝑦 ∣ ∃ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = ( +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)})
57 unab 4307 . . . . 5 ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑎 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)}) = {𝑎 ∣ (∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶) ∨ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶))}
58 eqeq1 2740 . . . . . . . 8 (𝑏 = 𝑎 → (𝑏 = ((𝐴 +s 𝑞) +s 𝐶) ↔ 𝑎 = ((𝐴 +s 𝑞) +s 𝐶)))
5958rexbidv 3178 . . . . . . 7 (𝑏 = 𝑎 → (∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)))
6059cbvabv 2811 . . . . . 6 {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶)} = {𝑎 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)}
6160uneq2i 4164 . . . . 5 ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶)}) = ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑎 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)})
62 rexun 4195 . . . . . . 7 (∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶) ↔ (∃𝑖 ∈ {𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)}𝑎 = (𝑖 +s 𝐶) ∨ ∃𝑖 ∈ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}𝑎 = (𝑖 +s 𝐶)))
63 eqeq1 2740 . . . . . . . . . . 11 (𝑓 = 𝑖 → (𝑓 = (𝑝 +s 𝐵) ↔ 𝑖 = (𝑝 +s 𝐵)))
6463rexbidv 3178 . . . . . . . . . 10 (𝑓 = 𝑖 → (∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵) ↔ ∃𝑝 ∈ ( R ‘𝐴)𝑖 = (𝑝 +s 𝐵)))
6564rexab 3699 . . . . . . . . 9 (∃𝑖 ∈ {𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)}𝑎 = (𝑖 +s 𝐶) ↔ ∃𝑖(∃𝑝 ∈ ( R ‘𝐴)𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)))
66 rexcom4 3287 . . . . . . . . . 10 (∃𝑝 ∈ ( R ‘𝐴)∃𝑖(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑖𝑝 ∈ ( R ‘𝐴)(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)))
67 ovex 7465 . . . . . . . . . . . 12 (𝑝 +s 𝐵) ∈ V
68 oveq1 7439 . . . . . . . . . . . . 13 (𝑖 = (𝑝 +s 𝐵) → (𝑖 +s 𝐶) = ((𝑝 +s 𝐵) +s 𝐶))
6968eqeq2d 2747 . . . . . . . . . . . 12 (𝑖 = (𝑝 +s 𝐵) → (𝑎 = (𝑖 +s 𝐶) ↔ 𝑎 = ((𝑝 +s 𝐵) +s 𝐶)))
7067, 69ceqsexv 3531 . . . . . . . . . . 11 (∃𝑖(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ 𝑎 = ((𝑝 +s 𝐵) +s 𝐶))
7170rexbii 3093 . . . . . . . . . 10 (∃𝑝 ∈ ( R ‘𝐴)∃𝑖(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶))
72 r19.41v 3188 . . . . . . . . . . 11 (∃𝑝 ∈ ( R ‘𝐴)(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ (∃𝑝 ∈ ( R ‘𝐴)𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)))
7372exbii 1847 . . . . . . . . . 10 (∃𝑖𝑝 ∈ ( R ‘𝐴)(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑖(∃𝑝 ∈ ( R ‘𝐴)𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)))
7466, 71, 733bitr3ri 302 . . . . . . . . 9 (∃𝑖(∃𝑝 ∈ ( R ‘𝐴)𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶))
7565, 74bitri 275 . . . . . . . 8 (∃𝑖 ∈ {𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)}𝑎 = (𝑖 +s 𝐶) ↔ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶))
76 eqeq1 2740 . . . . . . . . . . 11 (𝑔 = 𝑖 → (𝑔 = (𝐴 +s 𝑞) ↔ 𝑖 = (𝐴 +s 𝑞)))
7776rexbidv 3178 . . . . . . . . . 10 (𝑔 = 𝑖 → (∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝐴 +s 𝑞)))
7877rexab 3699 . . . . . . . . 9 (∃𝑖 ∈ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}𝑎 = (𝑖 +s 𝐶) ↔ ∃𝑖(∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)))
79 rexcom4 3287 . . . . . . . . . 10 (∃𝑞 ∈ ( R ‘𝐵)∃𝑖(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑖𝑞 ∈ ( R ‘𝐵)(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)))
80 ovex 7465 . . . . . . . . . . . 12 (𝐴 +s 𝑞) ∈ V
81 oveq1 7439 . . . . . . . . . . . . 13 (𝑖 = (𝐴 +s 𝑞) → (𝑖 +s 𝐶) = ((𝐴 +s 𝑞) +s 𝐶))
8281eqeq2d 2747 . . . . . . . . . . . 12 (𝑖 = (𝐴 +s 𝑞) → (𝑎 = (𝑖 +s 𝐶) ↔ 𝑎 = ((𝐴 +s 𝑞) +s 𝐶)))
8380, 82ceqsexv 3531 . . . . . . . . . . 11 (∃𝑖(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ 𝑎 = ((𝐴 +s 𝑞) +s 𝐶))
8483rexbii 3093 . . . . . . . . . 10 (∃𝑞 ∈ ( R ‘𝐵)∃𝑖(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶))
85 r19.41v 3188 . . . . . . . . . . 11 (∃𝑞 ∈ ( R ‘𝐵)(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ (∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)))
8685exbii 1847 . . . . . . . . . 10 (∃𝑖𝑞 ∈ ( R ‘𝐵)(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑖(∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)))
8779, 84, 863bitr3ri 302 . . . . . . . . 9 (∃𝑖(∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶))
8878, 87bitri 275 . . . . . . . 8 (∃𝑖 ∈ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}𝑎 = (𝑖 +s 𝐶) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶))
8975, 88orbi12i 914 . . . . . . 7 ((∃𝑖 ∈ {𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)}𝑎 = (𝑖 +s 𝐶) ∨ ∃𝑖 ∈ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}𝑎 = (𝑖 +s 𝐶)) ↔ (∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶) ∨ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)))
9062, 89bitri 275 . . . . . 6 (∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶) ↔ (∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶) ∨ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)))
9190abbii 2808 . . . . 5 {𝑎 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶)} = {𝑎 ∣ (∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶) ∨ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶))}
9257, 61, 913eqtr4ri 2775 . . . 4 {𝑎 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶)} = ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶)})
9392uneq1i 4163 . . 3 ({𝑎 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +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)})
9456, 93oveq12i 7444 . 2 (({𝑦 ∣ ∃ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = ( +s 𝐶)} ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = ((𝐴 +s 𝐵) +s 𝑛)}) |s ({𝑎 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶)} ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = ((𝐴 +s 𝐵) +s 𝑟)})) = ((({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)}))
9519, 94eqtrdi 2792 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 395  wo 847   = wceq 1539  wex 1778  wcel 2107  {cab 2713  wne 2939  wrex 3069  cun 3948  c0 4332  {csn 4625   class class class wbr 5142  cfv 6560  (class class class)co 7432   No csur 27685   <<s csslt 27826   |s cscut 27828   L cleft 27885   R cright 27886   +s cadds 27993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-1o 8507  df-2o 8508  df-nadd 8705  df-no 27688  df-slt 27689  df-bday 27690  df-sle 27791  df-sslt 27827  df-scut 27829  df-0s 27870  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec2 27983  df-adds 27994
This theorem is referenced by:  addsass  28039
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