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Theorem genpass 10033
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 10024 . . . . . . . . 9 ((𝐵P𝐶P) → (𝑡 ∈ (𝐵𝐹𝐶) ↔ ∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺)))
433adant1 1124 . . . . . . . 8 ((𝐴P𝐵P𝐶P) → (𝑡 ∈ (𝐵𝐹𝐶) ↔ ∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺)))
54anbi1d 615 . . . . . . 7 ((𝐴P𝐵P𝐶P) → ((𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
65exbidv 2002 . . . . . 6 ((𝐴P𝐵P𝐶P) → (∃𝑡(𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃𝑡(∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
7 df-rex 3067 . . . . . 6 (∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)))
8 ovex 6823 . . . . . . . . . . . . 13 (𝑔𝐺) ∈ V
98isseti 3361 . . . . . . . . . . . 12 𝑡 𝑡 = (𝑔𝐺)
109biantrur 520 . . . . . . . . . . 11 (𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ (∃𝑡 𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
11 19.41v 2029 . . . . . . . . . . 11 (∃𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝑡 𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
1210, 11bitr4i 267 . . . . . . . . . 10 (𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
1312rexbii 3189 . . . . . . . . 9 (∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝐶𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
14 rexcom4 3377 . . . . . . . . 9 (∃𝐶𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑡𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
1513, 14bitri 264 . . . . . . . 8 (∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
1615rexbii 3189 . . . . . . 7 (∃𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑔𝐵𝑡𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
17 rexcom4 3377 . . . . . . 7 (∃𝑔𝐵𝑡𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑡𝑔𝐵𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
18 oveq2 6801 . . . . . . . . . . . . . . 15 (𝑡 = (𝑔𝐺) → (𝑓𝐺𝑡) = (𝑓𝐺(𝑔𝐺)))
19 genpass.6 . . . . . . . . . . . . . . 15 ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺))
2018, 19syl6eqr 2823 . . . . . . . . . . . . . 14 (𝑡 = (𝑔𝐺) → (𝑓𝐺𝑡) = ((𝑓𝐺𝑔)𝐺))
2120eqeq2d 2781 . . . . . . . . . . . . 13 (𝑡 = (𝑔𝐺) → (𝑥 = (𝑓𝐺𝑡) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
2221pm5.32i 564 . . . . . . . . . . . 12 ((𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
2322rexbii 3189 . . . . . . . . . . 11 (∃𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
24 r19.41v 3237 . . . . . . . . . . 11 (∃𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
2523, 24bitr3i 266 . . . . . . . . . 10 (∃𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
2625rexbii 3189 . . . . . . . . 9 (∃𝑔𝐵𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑔𝐵 (∃𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
27 r19.41v 3237 . . . . . . . . 9 (∃𝑔𝐵 (∃𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
2826, 27bitri 264 . . . . . . . 8 (∃𝑔𝐵𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
2928exbii 1924 . . . . . . 7 (∃𝑡𝑔𝐵𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑡(∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
3016, 17, 293bitri 286 . . . . . 6 (∃𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
316, 7, 303bitr4g 303 . . . . 5 ((𝐴P𝐵P𝐶P) → (∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
3231rexbidv 3200 . . . 4 ((𝐴P𝐵P𝐶P) → (∃𝑓𝐴𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑓𝐴𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
33 genpass.5 . . . . . . 7 ((𝑓P𝑔P) → (𝑓𝐹𝑔) ∈ P)
3433caovcl 6975 . . . . . 6 ((𝐵P𝐶P) → (𝐵𝐹𝐶) ∈ P)
351, 2genpelv 10024 . . . . . 6 ((𝐴P ∧ (𝐵𝐹𝐶) ∈ P) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓𝐴𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
3634, 35sylan2 580 . . . . 5 ((𝐴P ∧ (𝐵P𝐶P)) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓𝐴𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
37363impb 1107 . . . 4 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓𝐴𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
3833caovcl 6975 . . . . . 6 ((𝐴P𝐵P) → (𝐴𝐹𝐵) ∈ P)
391, 2genpelv 10024 . . . . . 6 (((𝐴𝐹𝐵) ∈ P𝐶P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑡 ∈ (𝐴𝐹𝐵)∃𝐶 𝑥 = (𝑡𝐺)))
4038, 39stoic3 1849 . . . . 5 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑡 ∈ (𝐴𝐹𝐵)∃𝐶 𝑥 = (𝑡𝐺)))
411, 2genpelv 10024 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑡 ∈ (𝐴𝐹𝐵) ↔ ∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔)))
42413adant3 1126 . . . . . . . 8 ((𝐴P𝐵P𝐶P) → (𝑡 ∈ (𝐴𝐹𝐵) ↔ ∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔)))
4342anbi1d 615 . . . . . . 7 ((𝐴P𝐵P𝐶P) → ((𝑡 ∈ (𝐴𝐹𝐵) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ (∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺))))
4443exbidv 2002 . . . . . 6 ((𝐴P𝐵P𝐶P) → (∃𝑡(𝑡 ∈ (𝐴𝐹𝐵) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ ∃𝑡(∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺))))
45 df-rex 3067 . . . . . 6 (∃𝑡 ∈ (𝐴𝐹𝐵)∃𝐶 𝑥 = (𝑡𝐺) ↔ ∃𝑡(𝑡 ∈ (𝐴𝐹𝐵) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
46 19.41v 2029 . . . . . . . . . . 11 (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
47 oveq1 6800 . . . . . . . . . . . . . . 15 (𝑡 = (𝑓𝐺𝑔) → (𝑡𝐺) = ((𝑓𝐺𝑔)𝐺))
4847eqeq2d 2781 . . . . . . . . . . . . . 14 (𝑡 = (𝑓𝐺𝑔) → (𝑥 = (𝑡𝐺) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
4948rexbidv 3200 . . . . . . . . . . . . 13 (𝑡 = (𝑓𝐺𝑔) → (∃𝐶 𝑥 = (𝑡𝐺) ↔ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
5049pm5.32i 564 . . . . . . . . . . . 12 ((𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
5150exbii 1924 . . . . . . . . . . 11 (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
52 ovex 6823 . . . . . . . . . . . . 13 (𝑓𝐺𝑔) ∈ V
5352isseti 3361 . . . . . . . . . . . 12 𝑡 𝑡 = (𝑓𝐺𝑔)
5453biantrur 520 . . . . . . . . . . 11 (∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
5546, 51, 543bitr4ri 293 . . . . . . . . . 10 (∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
5655rexbii 3189 . . . . . . . . 9 (∃𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑔𝐵𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
57 rexcom4 3377 . . . . . . . . 9 (∃𝑔𝐵𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ ∃𝑡𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
5856, 57bitri 264 . . . . . . . 8 (∃𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
5958rexbii 3189 . . . . . . 7 (∃𝑓𝐴𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑓𝐴𝑡𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
60 rexcom4 3377 . . . . . . 7 (∃𝑓𝐴𝑡𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ ∃𝑡𝑓𝐴𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
61 r19.41vv 3239 . . . . . . . 8 (∃𝑓𝐴𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ (∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
6261exbii 1924 . . . . . . 7 (∃𝑡𝑓𝐴𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ ∃𝑡(∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
6359, 60, 623bitri 286 . . . . . 6 (∃𝑓𝐴𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
6444, 45, 633bitr4g 303 . . . . 5 ((𝐴P𝐵P𝐶P) → (∃𝑡 ∈ (𝐴𝐹𝐵)∃𝐶 𝑥 = (𝑡𝐺) ↔ ∃𝑓𝐴𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
6540, 64bitrd 268 . . . 4 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑓𝐴𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
6632, 37, 653bitr4rd 301 . . 3 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ 𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶))))
6766eqrdv 2769 . 2 ((𝐴P𝐵P𝐶P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
68 genpass.4 . . 3 dom 𝐹 = (P × P)
69 0npr 10016 . . 3 ¬ ∅ ∈ P
7068, 69ndmovass 6969 . 2 (¬ (𝐴P𝐵P𝐶P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
7167, 70pm2.61i 176 1 ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wrex 3062   × cxp 5247  dom cdm 5249  (class class class)co 6793  cmpt2 6795  Qcnq 9876  Pcnp 9883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-ni 9896  df-nq 9936  df-np 10005
This theorem is referenced by:  addasspr  10046  mulasspr  10048
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