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Theorem genpassl 6373
Description: Associativity of lower cuts. Lemma for genpassg 6375. (Contributed by Jim Kingdon, 11-Dec-2019.)
Hypotheses
Ref Expression
genpelvl.1 𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)
genpelvl.2 ((y Q z Q) → (y𝐺z) Q)
genpassg.4 dom 𝐹 = (P × P)
genpassg.5 ((f P g P) → (f𝐹g) P)
genpassg.6 ((f Q g Q Q) → ((f𝐺g)𝐺) = (f𝐺(g𝐺)))
Assertion
Ref Expression
genpassl ((A P B P 𝐶 P) → (1st ‘((A𝐹B)𝐹𝐶)) = (1st ‘(A𝐹(B𝐹𝐶))))
Distinct variable groups:   x,y,z,f,g,,w,v,A   x,B,y,z,f,g,,w,v   x,𝐺,y,z,f,g,,w,v   f,𝐹,g   𝐶,f,g,,v,w,x,y,z   ,𝐹,v,w,x,y,z

Proof of Theorem genpassl
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 prop 6323 . . . . . . . . 9 (A P → ⟨(1stA), (2ndA)⟩ P)
2 elprnql 6329 . . . . . . . . 9 ((⟨(1stA), (2ndA)⟩ P f (1stA)) → f Q)
31, 2sylan 267 . . . . . . . 8 ((A P f (1stA)) → f Q)
4 prop 6323 . . . . . . . . . . . . . . . 16 (B P → ⟨(1stB), (2ndB)⟩ P)
5 elprnql 6329 . . . . . . . . . . . . . . . 16 ((⟨(1stB), (2ndB)⟩ P g (1stB)) → g Q)
64, 5sylan 267 . . . . . . . . . . . . . . 15 ((B P g (1stB)) → g Q)
7 r19.41v 2440 . . . . . . . . . . . . . . . . 17 ( (1st𝐶)(𝑡 = (g𝐺) x = (f𝐺𝑡)) ↔ ( (1st𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡)))
8 prop 6323 . . . . . . . . . . . . . . . . . . . . 21 (𝐶 P → ⟨(1st𝐶), (2nd𝐶)⟩ P)
9 elprnql 6329 . . . . . . . . . . . . . . . . . . . . 21 ((⟨(1st𝐶), (2nd𝐶)⟩ P (1st𝐶)) → Q)
108, 9sylan 267 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 P (1st𝐶)) → Q)
11 oveq2 5440 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = (g𝐺) → (f𝐺𝑡) = (f𝐺(g𝐺)))
1211adantr 261 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 = (g𝐺) (f Q g Q Q)) → (f𝐺𝑡) = (f𝐺(g𝐺)))
13 genpassg.6 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((f Q g Q Q) → ((f𝐺g)𝐺) = (f𝐺(g𝐺)))
1413adantl 262 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 = (g𝐺) (f Q g Q Q)) → ((f𝐺g)𝐺) = (f𝐺(g𝐺)))
1512, 14eqtr4d 2053 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡 = (g𝐺) (f Q g Q Q)) → (f𝐺𝑡) = ((f𝐺g)𝐺))
1615eqeq2d 2029 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑡 = (g𝐺) (f Q g Q Q)) → (x = (f𝐺𝑡) ↔ x = ((f𝐺g)𝐺)))
1716expcom 109 . . . . . . . . . . . . . . . . . . . . . 22 ((f Q g Q Q) → (𝑡 = (g𝐺) → (x = (f𝐺𝑡) ↔ x = ((f𝐺g)𝐺))))
1817pm5.32d 426 . . . . . . . . . . . . . . . . . . . . 21 ((f Q g Q Q) → ((𝑡 = (g𝐺) x = (f𝐺𝑡)) ↔ (𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
19183expa 1088 . . . . . . . . . . . . . . . . . . . 20 (((f Q g Q) Q) → ((𝑡 = (g𝐺) x = (f𝐺𝑡)) ↔ (𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
2010, 19sylan2 270 . . . . . . . . . . . . . . . . . . 19 (((f Q g Q) (𝐶 P (1st𝐶))) → ((𝑡 = (g𝐺) x = (f𝐺𝑡)) ↔ (𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
2120anassrs 382 . . . . . . . . . . . . . . . . . 18 ((((f Q g Q) 𝐶 P) (1st𝐶)) → ((𝑡 = (g𝐺) x = (f𝐺𝑡)) ↔ (𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
2221rexbidva 2297 . . . . . . . . . . . . . . . . 17 (((f Q g Q) 𝐶 P) → ( (1st𝐶)(𝑡 = (g𝐺) x = (f𝐺𝑡)) ↔ (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
237, 22syl5rbbr 184 . . . . . . . . . . . . . . . 16 (((f Q g Q) 𝐶 P) → ( (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ ( (1st𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
2423an32s 490 . . . . . . . . . . . . . . 15 (((f Q 𝐶 P) g Q) → ( (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ ( (1st𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
256, 24sylan2 270 . . . . . . . . . . . . . 14 (((f Q 𝐶 P) (B P g (1stB))) → ( (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ ( (1st𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
2625anassrs 382 . . . . . . . . . . . . 13 ((((f Q 𝐶 P) B P) g (1stB)) → ( (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ ( (1st𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
2726rexbidva 2297 . . . . . . . . . . . 12 (((f Q 𝐶 P) B P) → (g (1stB) (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ g (1stB)( (1st𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
28 r19.41v 2440 . . . . . . . . . . . 12 (g (1stB)( (1st𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡)) ↔ (g (1stB) (1st𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡)))
2927, 28syl6bb 185 . . . . . . . . . . 11 (((f Q 𝐶 P) B P) → (g (1stB) (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ (g (1stB) (1st𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
3029an31s 492 . . . . . . . . . 10 (((B P 𝐶 P) f Q) → (g (1stB) (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ (g (1stB) (1st𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
3130exbidv 1684 . . . . . . . . 9 (((B P 𝐶 P) f Q) → (𝑡g (1stB) (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ 𝑡(g (1stB) (1st𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
32 genpelvl.2 . . . . . . . . . . . . . . . . . . . . . . . 24 ((y Q z Q) → (y𝐺z) Q)
3332caovcl 5574 . . . . . . . . . . . . . . . . . . . . . . 23 ((g Q Q) → (g𝐺) Q)
34 elisset 2541 . . . . . . . . . . . . . . . . . . . . . . 23 ((g𝐺) Q𝑡 𝑡 = (g𝐺))
3533, 34syl 14 . . . . . . . . . . . . . . . . . . . . . 22 ((g Q Q) → 𝑡 𝑡 = (g𝐺))
3635biantrurd 289 . . . . . . . . . . . . . . . . . . . . 21 ((g Q Q) → (x = ((f𝐺g)𝐺) ↔ (𝑡 𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
37 19.41v 1760 . . . . . . . . . . . . . . . . . . . . 21 (𝑡(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ (𝑡 𝑡 = (g𝐺) x = ((f𝐺g)𝐺)))
3836, 37syl6bbr 187 . . . . . . . . . . . . . . . . . . . 20 ((g Q Q) → (x = ((f𝐺g)𝐺) ↔ 𝑡(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
3910, 38sylan2 270 . . . . . . . . . . . . . . . . . . 19 ((g Q (𝐶 P (1st𝐶))) → (x = ((f𝐺g)𝐺) ↔ 𝑡(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
4039anassrs 382 . . . . . . . . . . . . . . . . . 18 (((g Q 𝐶 P) (1st𝐶)) → (x = ((f𝐺g)𝐺) ↔ 𝑡(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
4140rexbidva 2297 . . . . . . . . . . . . . . . . 17 ((g Q 𝐶 P) → ( (1st𝐶)x = ((f𝐺g)𝐺) ↔ (1st𝐶)𝑡(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
42 rexcom4 2550 . . . . . . . . . . . . . . . . 17 ( (1st𝐶)𝑡(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ 𝑡 (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)))
4341, 42syl6bb 185 . . . . . . . . . . . . . . . 16 ((g Q 𝐶 P) → ( (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡 (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
4443ancoms 255 . . . . . . . . . . . . . . 15 ((𝐶 P g Q) → ( (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡 (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
456, 44sylan2 270 . . . . . . . . . . . . . 14 ((𝐶 P (B P g (1stB))) → ( (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡 (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
4645anassrs 382 . . . . . . . . . . . . 13 (((𝐶 P B P) g (1stB)) → ( (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡 (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
4746rexbidva 2297 . . . . . . . . . . . 12 ((𝐶 P B P) → (g (1stB) (1st𝐶)x = ((f𝐺g)𝐺) ↔ g (1stB)𝑡 (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
4847ancoms 255 . . . . . . . . . . 11 ((B P 𝐶 P) → (g (1stB) (1st𝐶)x = ((f𝐺g)𝐺) ↔ g (1stB)𝑡 (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
49 rexcom4 2550 . . . . . . . . . . 11 (g (1stB)𝑡 (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ 𝑡g (1stB) (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)))
5048, 49syl6bb 185 . . . . . . . . . 10 ((B P 𝐶 P) → (g (1stB) (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡g (1stB) (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
5150adantr 261 . . . . . . . . 9 (((B P 𝐶 P) f Q) → (g (1stB) (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡g (1stB) (1st𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
52 df-rex 2286 . . . . . . . . . . 11 (𝑡 (1st ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ 𝑡(𝑡 (1st ‘(B𝐹𝐶)) x = (f𝐺𝑡)))
53 genpelvl.1 . . . . . . . . . . . . . 14 𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)
5453, 32genpelvl 6360 . . . . . . . . . . . . 13 ((B P 𝐶 P) → (𝑡 (1st ‘(B𝐹𝐶)) ↔ g (1stB) (1st𝐶)𝑡 = (g𝐺)))
5554anbi1d 441 . . . . . . . . . . . 12 ((B P 𝐶 P) → ((𝑡 (1st ‘(B𝐹𝐶)) x = (f𝐺𝑡)) ↔ (g (1stB) (1st𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
5655exbidv 1684 . . . . . . . . . . 11 ((B P 𝐶 P) → (𝑡(𝑡 (1st ‘(B𝐹𝐶)) x = (f𝐺𝑡)) ↔ 𝑡(g (1stB) (1st𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
5752, 56syl5bb 181 . . . . . . . . . 10 ((B P 𝐶 P) → (𝑡 (1st ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ 𝑡(g (1stB) (1st𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
5857adantr 261 . . . . . . . . 9 (((B P 𝐶 P) f Q) → (𝑡 (1st ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ 𝑡(g (1stB) (1st𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
5931, 51, 583bitr4rd 210 . . . . . . . 8 (((B P 𝐶 P) f Q) → (𝑡 (1st ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ g (1stB) (1st𝐶)x = ((f𝐺g)𝐺)))
603, 59sylan2 270 . . . . . . 7 (((B P 𝐶 P) (A P f (1stA))) → (𝑡 (1st ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ g (1stB) (1st𝐶)x = ((f𝐺g)𝐺)))
6160anassrs 382 . . . . . 6 ((((B P 𝐶 P) A P) f (1stA)) → (𝑡 (1st ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ g (1stB) (1st𝐶)x = ((f𝐺g)𝐺)))
6261rexbidva 2297 . . . . 5 (((B P 𝐶 P) A P) → (f (1stA)𝑡 (1st ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ f (1stA)g (1stB) (1st𝐶)x = ((f𝐺g)𝐺)))
6362ancoms 255 . . . 4 ((A P (B P 𝐶 P)) → (f (1stA)𝑡 (1st ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ f (1stA)g (1stB) (1st𝐶)x = ((f𝐺g)𝐺)))
64633impb 1084 . . 3 ((A P B P 𝐶 P) → (f (1stA)𝑡 (1st ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ f (1stA)g (1stB) (1st𝐶)x = ((f𝐺g)𝐺)))
65 genpassg.5 . . . . . 6 ((f P g P) → (f𝐹g) P)
6665caovcl 5574 . . . . 5 ((B P 𝐶 P) → (B𝐹𝐶) P)
6753, 32genpelvl 6360 . . . . 5 ((A P (B𝐹𝐶) P) → (x (1st ‘(A𝐹(B𝐹𝐶))) ↔ f (1stA)𝑡 (1st ‘(B𝐹𝐶))x = (f𝐺𝑡)))
6866, 67sylan2 270 . . . 4 ((A P (B P 𝐶 P)) → (x (1st ‘(A𝐹(B𝐹𝐶))) ↔ f (1stA)𝑡 (1st ‘(B𝐹𝐶))x = (f𝐺𝑡)))
69683impb 1084 . . 3 ((A P B P 𝐶 P) → (x (1st ‘(A𝐹(B𝐹𝐶))) ↔ f (1stA)𝑡 (1st ‘(B𝐹𝐶))x = (f𝐺𝑡)))
70 df-rex 2286 . . . . 5 (𝑡 (1st ‘(A𝐹B)) (1st𝐶)x = (𝑡𝐺) ↔ 𝑡(𝑡 (1st ‘(A𝐹B)) (1st𝐶)x = (𝑡𝐺)))
7153, 32genpelvl 6360 . . . . . . . 8 ((A P B P) → (𝑡 (1st ‘(A𝐹B)) ↔ f (1stA)g (1stB)𝑡 = (f𝐺g)))
72713adant3 910 . . . . . . 7 ((A P B P 𝐶 P) → (𝑡 (1st ‘(A𝐹B)) ↔ f (1stA)g (1stB)𝑡 = (f𝐺g)))
7372anbi1d 441 . . . . . 6 ((A P B P 𝐶 P) → ((𝑡 (1st ‘(A𝐹B)) (1st𝐶)x = (𝑡𝐺)) ↔ (f (1stA)g (1stB)𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺))))
7473exbidv 1684 . . . . 5 ((A P B P 𝐶 P) → (𝑡(𝑡 (1st ‘(A𝐹B)) (1st𝐶)x = (𝑡𝐺)) ↔ 𝑡(f (1stA)g (1stB)𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺))))
7570, 74syl5bb 181 . . . 4 ((A P B P 𝐶 P) → (𝑡 (1st ‘(A𝐹B)) (1st𝐶)x = (𝑡𝐺) ↔ 𝑡(f (1stA)g (1stB)𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺))))
7665caovcl 5574 . . . . . 6 ((A P B P) → (A𝐹B) P)
7753, 32genpelvl 6360 . . . . . 6 (((A𝐹B) P 𝐶 P) → (x (1st ‘((A𝐹B)𝐹𝐶)) ↔ 𝑡 (1st ‘(A𝐹B)) (1st𝐶)x = (𝑡𝐺)))
7876, 77sylan 267 . . . . 5 (((A P B P) 𝐶 P) → (x (1st ‘((A𝐹B)𝐹𝐶)) ↔ 𝑡 (1st ‘(A𝐹B)) (1st𝐶)x = (𝑡𝐺)))
79783impa 1083 . . . 4 ((A P B P 𝐶 P) → (x (1st ‘((A𝐹B)𝐹𝐶)) ↔ 𝑡 (1st ‘(A𝐹B)) (1st𝐶)x = (𝑡𝐺)))
8032caovcl 5574 . . . . . . . . . . . . . . . . . . 19 ((f Q g Q) → (f𝐺g) Q)
81 elisset 2541 . . . . . . . . . . . . . . . . . . 19 ((f𝐺g) Q𝑡 𝑡 = (f𝐺g))
8280, 81syl 14 . . . . . . . . . . . . . . . . . 18 ((f Q g Q) → 𝑡 𝑡 = (f𝐺g))
8382biantrurd 289 . . . . . . . . . . . . . . . . 17 ((f Q g Q) → ( (1st𝐶)x = ((f𝐺g)𝐺) ↔ (𝑡 𝑡 = (f𝐺g) (1st𝐶)x = ((f𝐺g)𝐺))))
84 oveq1 5439 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = (f𝐺g) → (𝑡𝐺) = ((f𝐺g)𝐺))
8584eqeq2d 2029 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = (f𝐺g) → (x = (𝑡𝐺) ↔ x = ((f𝐺g)𝐺)))
8685rexbidv 2301 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = (f𝐺g) → ( (1st𝐶)x = (𝑡𝐺) ↔ (1st𝐶)x = ((f𝐺g)𝐺)))
8786pm5.32i 430 . . . . . . . . . . . . . . . . . . 19 ((𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)) ↔ (𝑡 = (f𝐺g) (1st𝐶)x = ((f𝐺g)𝐺)))
8887exbii 1474 . . . . . . . . . . . . . . . . . 18 (𝑡(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)) ↔ 𝑡(𝑡 = (f𝐺g) (1st𝐶)x = ((f𝐺g)𝐺)))
89 19.41v 1760 . . . . . . . . . . . . . . . . . 18 (𝑡(𝑡 = (f𝐺g) (1st𝐶)x = ((f𝐺g)𝐺)) ↔ (𝑡 𝑡 = (f𝐺g) (1st𝐶)x = ((f𝐺g)𝐺)))
9088, 89bitri 173 . . . . . . . . . . . . . . . . 17 (𝑡(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)) ↔ (𝑡 𝑡 = (f𝐺g) (1st𝐶)x = ((f𝐺g)𝐺)))
9183, 90syl6bbr 187 . . . . . . . . . . . . . . . 16 ((f Q g Q) → ( (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺))))
926, 91sylan2 270 . . . . . . . . . . . . . . 15 ((f Q (B P g (1stB))) → ( (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺))))
9392anassrs 382 . . . . . . . . . . . . . 14 (((f Q B P) g (1stB)) → ( (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺))))
9493rexbidva 2297 . . . . . . . . . . . . 13 ((f Q B P) → (g (1stB) (1st𝐶)x = ((f𝐺g)𝐺) ↔ g (1stB)𝑡(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺))))
95 rexcom4 2550 . . . . . . . . . . . . 13 (g (1stB)𝑡(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)) ↔ 𝑡g (1stB)(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)))
9694, 95syl6bb 185 . . . . . . . . . . . 12 ((f Q B P) → (g (1stB) (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡g (1stB)(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺))))
9796ancoms 255 . . . . . . . . . . 11 ((B P f Q) → (g (1stB) (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡g (1stB)(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺))))
983, 97sylan2 270 . . . . . . . . . 10 ((B P (A P f (1stA))) → (g (1stB) (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡g (1stB)(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺))))
9998anassrs 382 . . . . . . . . 9 (((B P A P) f (1stA)) → (g (1stB) (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡g (1stB)(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺))))
10099rexbidva 2297 . . . . . . . 8 ((B P A P) → (f (1stA)g (1stB) (1st𝐶)x = ((f𝐺g)𝐺) ↔ f (1stA)𝑡g (1stB)(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺))))
101 rexcom4 2550 . . . . . . . 8 (f (1stA)𝑡g (1stB)(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)) ↔ 𝑡f (1stA)g (1stB)(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)))
102100, 101syl6bb 185 . . . . . . 7 ((B P A P) → (f (1stA)g (1stB) (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡f (1stA)g (1stB)(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺))))
103 r19.41v 2440 . . . . . . . . . 10 (g (1stB)(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)) ↔ (g (1stB)𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)))
104103rexbii 2305 . . . . . . . . 9 (f (1stA)g (1stB)(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)) ↔ f (1stA)(g (1stB)𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)))
105 r19.41v 2440 . . . . . . . . 9 (f (1stA)(g (1stB)𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)) ↔ (f (1stA)g (1stB)𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)))
106104, 105bitri 173 . . . . . . . 8 (f (1stA)g (1stB)(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)) ↔ (f (1stA)g (1stB)𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)))
107106exbii 1474 . . . . . . 7 (𝑡f (1stA)g (1stB)(𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)) ↔ 𝑡(f (1stA)g (1stB)𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺)))
108102, 107syl6bb 185 . . . . . 6 ((B P A P) → (f (1stA)g (1stB) (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡(f (1stA)g (1stB)𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺))))
109108ancoms 255 . . . . 5 ((A P B P) → (f (1stA)g (1stB) (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡(f (1stA)g (1stB)𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺))))
1101093adant3 910 . . . 4 ((A P B P 𝐶 P) → (f (1stA)g (1stB) (1st𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡(f (1stA)g (1stB)𝑡 = (f𝐺g) (1st𝐶)x = (𝑡𝐺))))
11175, 79, 1103bitr4d 209 . . 3 ((A P B P 𝐶 P) → (x (1st ‘((A𝐹B)𝐹𝐶)) ↔ f (1stA)g (1stB) (1st𝐶)x = ((f𝐺g)𝐺)))
11264, 69, 1113bitr4rd 210 . 2 ((A P B P 𝐶 P) → (x (1st ‘((A𝐹B)𝐹𝐶)) ↔ x (1st ‘(A𝐹(B𝐹𝐶)))))
113112eqrdv 2016 1 ((A P B P 𝐶 P) → (1st ‘((A𝐹B)𝐹𝐶)) = (1st ‘(A𝐹(B𝐹𝐶))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 871   = wceq 1226  wex 1358   wcel 1370  wrex 2281  {crab 2284  cop 3349   × cxp 4266  dom cdm 4268  cfv 4825  (class class class)co 5432  cmpt2 5434  1st c1st 5684  2nd c2nd 5685  Qcnq 6134  Pcnp 6145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200  ax-iinf 4234
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-iom 4237  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-1st 5686  df-2nd 5687  df-qs 6019  df-ni 6158  df-nqqs 6201  df-inp 6314
This theorem is referenced by:  genpassg  6375
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