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Theorem genpass 11078
Description: Associativity of an operation on reals. (Contributed by NM, 18-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
genp.1 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
genp.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
genpass.4 dom 𝐹 = (P × P)
genpass.5 ((𝑓P𝑔P) → (𝑓𝐹𝑔) ∈ P)
genpass.6 ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺))
Assertion
Ref Expression
genpass ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑓,𝑔,,𝐴   𝑥,𝐵,𝑦,𝑧,𝑓,𝑔,   𝑥,𝑤,𝑣,𝐺,𝑦,𝑧,𝑓,𝑔,   𝑓,𝐹,𝑔   𝐶,𝑓,𝑔,,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧,
Allowed substitution hints:   𝐴(𝑤,𝑣)   𝐵(𝑤,𝑣)   𝐶(𝑤,𝑣)   𝐹(𝑤,𝑣)

Proof of Theorem genpass
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 genp.1 . . . . . . . . . 10 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
2 genp.2 . . . . . . . . . 10 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
31, 2genpelv 11069 . . . . . . . . 9 ((𝐵P𝐶P) → (𝑡 ∈ (𝐵𝐹𝐶) ↔ ∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺)))
433adant1 1130 . . . . . . . 8 ((𝐴P𝐵P𝐶P) → (𝑡 ∈ (𝐵𝐹𝐶) ↔ ∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺)))
54anbi1d 630 . . . . . . 7 ((𝐴P𝐵P𝐶P) → ((𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
65exbidv 1920 . . . . . 6 ((𝐴P𝐵P𝐶P) → (∃𝑡(𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃𝑡(∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
7 df-rex 3077 . . . . . 6 (∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)))
8 ovex 7481 . . . . . . . . . . . . 13 (𝑔𝐺) ∈ V
98isseti 3506 . . . . . . . . . . . 12 𝑡 𝑡 = (𝑔𝐺)
109biantrur 530 . . . . . . . . . . 11 (𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ (∃𝑡 𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
11 19.41v 1949 . . . . . . . . . . 11 (∃𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝑡 𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
1210, 11bitr4i 278 . . . . . . . . . 10 (𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
1312rexbii 3100 . . . . . . . . 9 (∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝐶𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
14 rexcom4 3294 . . . . . . . . 9 (∃𝐶𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑡𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
1513, 14bitri 275 . . . . . . . 8 (∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
1615rexbii 3100 . . . . . . 7 (∃𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑔𝐵𝑡𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
17 rexcom4 3294 . . . . . . 7 (∃𝑔𝐵𝑡𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑡𝑔𝐵𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
18 oveq2 7456 . . . . . . . . . . . . . . 15 (𝑡 = (𝑔𝐺) → (𝑓𝐺𝑡) = (𝑓𝐺(𝑔𝐺)))
19 genpass.6 . . . . . . . . . . . . . . 15 ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺))
2018, 19eqtr4di 2798 . . . . . . . . . . . . . 14 (𝑡 = (𝑔𝐺) → (𝑓𝐺𝑡) = ((𝑓𝐺𝑔)𝐺))
2120eqeq2d 2751 . . . . . . . . . . . . 13 (𝑡 = (𝑔𝐺) → (𝑥 = (𝑓𝐺𝑡) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
2221pm5.32i 574 . . . . . . . . . . . 12 ((𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
2322rexbii 3100 . . . . . . . . . . 11 (∃𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
24 r19.41v 3195 . . . . . . . . . . 11 (∃𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
2523, 24bitr3i 277 . . . . . . . . . 10 (∃𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
2625rexbii 3100 . . . . . . . . 9 (∃𝑔𝐵𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑔𝐵 (∃𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
27 r19.41v 3195 . . . . . . . . 9 (∃𝑔𝐵 (∃𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
2826, 27bitri 275 . . . . . . . 8 (∃𝑔𝐵𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
2928exbii 1846 . . . . . . 7 (∃𝑡𝑔𝐵𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑡(∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
3016, 17, 293bitri 297 . . . . . 6 (∃𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
316, 7, 303bitr4g 314 . . . . 5 ((𝐴P𝐵P𝐶P) → (∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
3231rexbidv 3185 . . . 4 ((𝐴P𝐵P𝐶P) → (∃𝑓𝐴𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑓𝐴𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
33 genpass.5 . . . . . . 7 ((𝑓P𝑔P) → (𝑓𝐹𝑔) ∈ P)
3433caovcl 7644 . . . . . 6 ((𝐵P𝐶P) → (𝐵𝐹𝐶) ∈ P)
351, 2genpelv 11069 . . . . . 6 ((𝐴P ∧ (𝐵𝐹𝐶) ∈ P) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓𝐴𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
3634, 35sylan2 592 . . . . 5 ((𝐴P ∧ (𝐵P𝐶P)) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓𝐴𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
37363impb 1115 . . . 4 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓𝐴𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
3833caovcl 7644 . . . . . 6 ((𝐴P𝐵P) → (𝐴𝐹𝐵) ∈ P)
391, 2genpelv 11069 . . . . . 6 (((𝐴𝐹𝐵) ∈ P𝐶P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑡 ∈ (𝐴𝐹𝐵)∃𝐶 𝑥 = (𝑡𝐺)))
4038, 39stoic3 1774 . . . . 5 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑡 ∈ (𝐴𝐹𝐵)∃𝐶 𝑥 = (𝑡𝐺)))
411, 2genpelv 11069 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑡 ∈ (𝐴𝐹𝐵) ↔ ∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔)))
42413adant3 1132 . . . . . . . 8 ((𝐴P𝐵P𝐶P) → (𝑡 ∈ (𝐴𝐹𝐵) ↔ ∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔)))
4342anbi1d 630 . . . . . . 7 ((𝐴P𝐵P𝐶P) → ((𝑡 ∈ (𝐴𝐹𝐵) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ (∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺))))
4443exbidv 1920 . . . . . 6 ((𝐴P𝐵P𝐶P) → (∃𝑡(𝑡 ∈ (𝐴𝐹𝐵) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ ∃𝑡(∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺))))
45 df-rex 3077 . . . . . 6 (∃𝑡 ∈ (𝐴𝐹𝐵)∃𝐶 𝑥 = (𝑡𝐺) ↔ ∃𝑡(𝑡 ∈ (𝐴𝐹𝐵) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
46 19.41v 1949 . . . . . . . . . . 11 (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
47 oveq1 7455 . . . . . . . . . . . . . . 15 (𝑡 = (𝑓𝐺𝑔) → (𝑡𝐺) = ((𝑓𝐺𝑔)𝐺))
4847eqeq2d 2751 . . . . . . . . . . . . . 14 (𝑡 = (𝑓𝐺𝑔) → (𝑥 = (𝑡𝐺) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
4948rexbidv 3185 . . . . . . . . . . . . 13 (𝑡 = (𝑓𝐺𝑔) → (∃𝐶 𝑥 = (𝑡𝐺) ↔ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
5049pm5.32i 574 . . . . . . . . . . . 12 ((𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
5150exbii 1846 . . . . . . . . . . 11 (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
52 ovex 7481 . . . . . . . . . . . . 13 (𝑓𝐺𝑔) ∈ V
5352isseti 3506 . . . . . . . . . . . 12 𝑡 𝑡 = (𝑓𝐺𝑔)
5453biantrur 530 . . . . . . . . . . 11 (∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
5546, 51, 543bitr4ri 304 . . . . . . . . . 10 (∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
5655rexbii 3100 . . . . . . . . 9 (∃𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑔𝐵𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
57 rexcom4 3294 . . . . . . . . 9 (∃𝑔𝐵𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ ∃𝑡𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
5856, 57bitri 275 . . . . . . . 8 (∃𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
5958rexbii 3100 . . . . . . 7 (∃𝑓𝐴𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑓𝐴𝑡𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
60 rexcom4 3294 . . . . . . 7 (∃𝑓𝐴𝑡𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ ∃𝑡𝑓𝐴𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
61 r19.41vv 3233 . . . . . . . 8 (∃𝑓𝐴𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ (∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
6261exbii 1846 . . . . . . 7 (∃𝑡𝑓𝐴𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ ∃𝑡(∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
6359, 60, 623bitri 297 . . . . . 6 (∃𝑓𝐴𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
6444, 45, 633bitr4g 314 . . . . 5 ((𝐴P𝐵P𝐶P) → (∃𝑡 ∈ (𝐴𝐹𝐵)∃𝐶 𝑥 = (𝑡𝐺) ↔ ∃𝑓𝐴𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
6540, 64bitrd 279 . . . 4 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑓𝐴𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
6632, 37, 653bitr4rd 312 . . 3 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ 𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶))))
6766eqrdv 2738 . 2 ((𝐴P𝐵P𝐶P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
68 genpass.4 . . 3 dom 𝐹 = (P × P)
69 0npr 11061 . . 3 ¬ ∅ ∈ P
7068, 69ndmovass 7638 . 2 (¬ (𝐴P𝐵P𝐶P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
7167, 70pm2.61i 182 1 ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wex 1777  wcel 2108  {cab 2717  wrex 3076   × cxp 5698  dom cdm 5700  (class class class)co 7448  cmpo 7450  Qcnq 10921  Pcnp 10928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-ni 10941  df-nq 10981  df-np 11050
This theorem is referenced by:  addasspr  11091  mulasspr  11093
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