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Theorem genpass 9775
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 9766 . . . . . . . . 9 ((𝐵P𝐶P) → (𝑡 ∈ (𝐵𝐹𝐶) ↔ ∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺)))
433adant1 1077 . . . . . . . 8 ((𝐴P𝐵P𝐶P) → (𝑡 ∈ (𝐵𝐹𝐶) ↔ ∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺)))
54anbi1d 740 . . . . . . 7 ((𝐴P𝐵P𝐶P) → ((𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
65exbidv 1847 . . . . . 6 ((𝐴P𝐵P𝐶P) → (∃𝑡(𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃𝑡(∃𝑔𝐵𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
7 df-rex 2913 . . . . . 6 (∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)))
8 ovex 6632 . . . . . . . . . . . . 13 (𝑔𝐺) ∈ V
98isseti 3195 . . . . . . . . . . . 12 𝑡 𝑡 = (𝑔𝐺)
109biantrur 527 . . . . . . . . . . 11 (𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ (∃𝑡 𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
11 19.41v 1911 . . . . . . . . . . 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 3034 . . . . . . . . 9 (∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝐶𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
14 rexcom4 3211 . . . . . . . . 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 3034 . . . . . . 7 (∃𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑔𝐵𝑡𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
17 rexcom4 3211 . . . . . . 7 (∃𝑔𝐵𝑡𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑡𝑔𝐵𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
18 oveq2 6612 . . . . . . . . . . . . . . 15 (𝑡 = (𝑔𝐺) → (𝑓𝐺𝑡) = (𝑓𝐺(𝑔𝐺)))
19 genpass.6 . . . . . . . . . . . . . . 15 ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺))
2018, 19syl6eqr 2673 . . . . . . . . . . . . . 14 (𝑡 = (𝑔𝐺) → (𝑓𝐺𝑡) = ((𝑓𝐺𝑔)𝐺))
2120eqeq2d 2631 . . . . . . . . . . . . 13 (𝑡 = (𝑔𝐺) → (𝑥 = (𝑓𝐺𝑡) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
2221pm5.32i 668 . . . . . . . . . . . 12 ((𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
2322rexbii 3034 . . . . . . . . . . 11 (∃𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
24 r19.41v 3081 . . . . . . . . . . 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 3034 . . . . . . . . 9 (∃𝑔𝐵𝐶 (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑔𝐵 (∃𝐶 𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
27 r19.41v 3081 . . . . . . . . 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 1771 . . . . . . 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 3045 . . . 4 ((𝐴P𝐵P𝐶P) → (∃𝑓𝐴𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑓𝐴𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
33 genpass.5 . . . . . . 7 ((𝑓P𝑔P) → (𝑓𝐹𝑔) ∈ P)
3433caovcl 6781 . . . . . 6 ((𝐵P𝐶P) → (𝐵𝐹𝐶) ∈ P)
351, 2genpelv 9766 . . . . . 6 ((𝐴P ∧ (𝐵𝐹𝐶) ∈ P) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓𝐴𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
3634, 35sylan2 491 . . . . 5 ((𝐴P ∧ (𝐵P𝐶P)) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓𝐴𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
37363impb 1257 . . . 4 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓𝐴𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
3833caovcl 6781 . . . . . 6 ((𝐴P𝐵P) → (𝐴𝐹𝐵) ∈ P)
391, 2genpelv 9766 . . . . . 6 (((𝐴𝐹𝐵) ∈ P𝐶P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑡 ∈ (𝐴𝐹𝐵)∃𝐶 𝑥 = (𝑡𝐺)))
4038, 39stoic3 1698 . . . . 5 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑡 ∈ (𝐴𝐹𝐵)∃𝐶 𝑥 = (𝑡𝐺)))
411, 2genpelv 9766 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑡 ∈ (𝐴𝐹𝐵) ↔ ∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔)))
42413adant3 1079 . . . . . . . 8 ((𝐴P𝐵P𝐶P) → (𝑡 ∈ (𝐴𝐹𝐵) ↔ ∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔)))
4342anbi1d 740 . . . . . . 7 ((𝐴P𝐵P𝐶P) → ((𝑡 ∈ (𝐴𝐹𝐵) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ (∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺))))
4443exbidv 1847 . . . . . 6 ((𝐴P𝐵P𝐶P) → (∃𝑡(𝑡 ∈ (𝐴𝐹𝐵) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ ∃𝑡(∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺))))
45 df-rex 2913 . . . . . 6 (∃𝑡 ∈ (𝐴𝐹𝐵)∃𝐶 𝑥 = (𝑡𝐺) ↔ ∃𝑡(𝑡 ∈ (𝐴𝐹𝐵) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
46 19.41v 1911 . . . . . . . . . . 11 (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
47 oveq1 6611 . . . . . . . . . . . . . . 15 (𝑡 = (𝑓𝐺𝑔) → (𝑡𝐺) = ((𝑓𝐺𝑔)𝐺))
4847eqeq2d 2631 . . . . . . . . . . . . . 14 (𝑡 = (𝑓𝐺𝑔) → (𝑥 = (𝑡𝐺) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
4948rexbidv 3045 . . . . . . . . . . . . 13 (𝑡 = (𝑓𝐺𝑔) → (∃𝐶 𝑥 = (𝑡𝐺) ↔ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
5049pm5.32i 668 . . . . . . . . . . . 12 ((𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
5150exbii 1771 . . . . . . . . . . 11 (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺)))
52 ovex 6632 . . . . . . . . . . . . 13 (𝑓𝐺𝑔) ∈ V
5352isseti 3195 . . . . . . . . . . . 12 𝑡 𝑡 = (𝑓𝐺𝑔)
5453biantrur 527 . . . . . . . . . . 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 3034 . . . . . . . . 9 (∃𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑔𝐵𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
57 rexcom4 3211 . . . . . . . . 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 3034 . . . . . . 7 (∃𝑓𝐴𝑔𝐵𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑓𝐴𝑡𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
60 rexcom4 3211 . . . . . . 7 (∃𝑓𝐴𝑡𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ ∃𝑡𝑓𝐴𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
61 r19.41vv 3083 . . . . . . . 8 (∃𝑓𝐴𝑔𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)) ↔ (∃𝑓𝐴𝑔𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃𝐶 𝑥 = (𝑡𝐺)))
6261exbii 1771 . . . . . . 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 2619 . 2 ((𝐴P𝐵P𝐶P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
68 genpass.4 . . 3 dom 𝐹 = (P × P)
69 0npr 9758 . . 3 ¬ ∅ ∈ P
7068, 69ndmovass 6775 . 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 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wrex 2908   × cxp 5072  dom cdm 5074  (class class class)co 6604  cmpt2 6606  Qcnq 9618  Pcnp 9625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-ni 9638  df-nq 9678  df-np 9747
This theorem is referenced by:  addasspr  9788  mulasspr  9790
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