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Theorem genpassu 6508
Description: Associativity of upper cuts. Lemma for genpassg 6509. (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
genpassu ((A P B P 𝐶 P) → (2nd ‘((A𝐹B)𝐹𝐶)) = (2nd ‘(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 genpassu
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 prop 6458 . . . . . . . . 9 (A P → ⟨(1stA), (2ndA)⟩ P)
2 elprnqu 6465 . . . . . . . . 9 ((⟨(1stA), (2ndA)⟩ P f (2ndA)) → f Q)
31, 2sylan 267 . . . . . . . 8 ((A P f (2ndA)) → f Q)
4 prop 6458 . . . . . . . . . . . . . . . 16 (B P → ⟨(1stB), (2ndB)⟩ P)
5 elprnqu 6465 . . . . . . . . . . . . . . . 16 ((⟨(1stB), (2ndB)⟩ P g (2ndB)) → g Q)
64, 5sylan 267 . . . . . . . . . . . . . . 15 ((B P g (2ndB)) → g Q)
7 r19.41v 2460 . . . . . . . . . . . . . . . . 17 ( (2nd𝐶)(𝑡 = (g𝐺) x = (f𝐺𝑡)) ↔ ( (2nd𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡)))
8 prop 6458 . . . . . . . . . . . . . . . . . . . . 21 (𝐶 P → ⟨(1st𝐶), (2nd𝐶)⟩ P)
9 elprnqu 6465 . . . . . . . . . . . . . . . . . . . . 21 ((⟨(1st𝐶), (2nd𝐶)⟩ P (2nd𝐶)) → Q)
108, 9sylan 267 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 P (2nd𝐶)) → Q)
11 oveq2 5463 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2072 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡 = (g𝐺) (f Q g Q Q)) → (f𝐺𝑡) = ((f𝐺g)𝐺))
1615eqeq2d 2048 . . . . . . . . . . . . . . . . . . . . . . 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 423 . . . . . . . . . . . . . . . . . . . . 21 ((f Q g Q Q) → ((𝑡 = (g𝐺) x = (f𝐺𝑡)) ↔ (𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
19183expa 1103 . . . . . . . . . . . . . . . . . . . 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 (2nd𝐶))) → ((𝑡 = (g𝐺) x = (f𝐺𝑡)) ↔ (𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
2120anassrs 380 . . . . . . . . . . . . . . . . . 18 ((((f Q g Q) 𝐶 P) (2nd𝐶)) → ((𝑡 = (g𝐺) x = (f𝐺𝑡)) ↔ (𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
2221rexbidva 2317 . . . . . . . . . . . . . . . . 17 (((f Q g Q) 𝐶 P) → ( (2nd𝐶)(𝑡 = (g𝐺) x = (f𝐺𝑡)) ↔ (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
237, 22syl5rbbr 184 . . . . . . . . . . . . . . . 16 (((f Q g Q) 𝐶 P) → ( (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ ( (2nd𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
2423an32s 502 . . . . . . . . . . . . . . 15 (((f Q 𝐶 P) g Q) → ( (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ ( (2nd𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
256, 24sylan2 270 . . . . . . . . . . . . . 14 (((f Q 𝐶 P) (B P g (2ndB))) → ( (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ ( (2nd𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
2625anassrs 380 . . . . . . . . . . . . 13 ((((f Q 𝐶 P) B P) g (2ndB)) → ( (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ ( (2nd𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
2726rexbidva 2317 . . . . . . . . . . . 12 (((f Q 𝐶 P) B P) → (g (2ndB) (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ g (2ndB)( (2nd𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
28 r19.41v 2460 . . . . . . . . . . . 12 (g (2ndB)( (2nd𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡)) ↔ (g (2ndB) (2nd𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡)))
2927, 28syl6bb 185 . . . . . . . . . . 11 (((f Q 𝐶 P) B P) → (g (2ndB) (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ (g (2ndB) (2nd𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
3029an31s 504 . . . . . . . . . 10 (((B P 𝐶 P) f Q) → (g (2ndB) (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ (g (2ndB) (2nd𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
3130exbidv 1703 . . . . . . . . 9 (((B P 𝐶 P) f Q) → (𝑡g (2ndB) (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ 𝑡(g (2ndB) (2nd𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
32 genpelvl.2 . . . . . . . . . . . . . . . . . . . . . . . 24 ((y Q z Q) → (y𝐺z) Q)
3332caovcl 5597 . . . . . . . . . . . . . . . . . . . . . . 23 ((g Q Q) → (g𝐺) Q)
34 elisset 2562 . . . . . . . . . . . . . . . . . . . . . . 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 1779 . . . . . . . . . . . . . . . . . . . . 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 (2nd𝐶))) → (x = ((f𝐺g)𝐺) ↔ 𝑡(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
4039anassrs 380 . . . . . . . . . . . . . . . . . 18 (((g Q 𝐶 P) (2nd𝐶)) → (x = ((f𝐺g)𝐺) ↔ 𝑡(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
4140rexbidva 2317 . . . . . . . . . . . . . . . . 17 ((g Q 𝐶 P) → ( (2nd𝐶)x = ((f𝐺g)𝐺) ↔ (2nd𝐶)𝑡(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
42 rexcom4 2571 . . . . . . . . . . . . . . . . 17 ( (2nd𝐶)𝑡(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ 𝑡 (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)))
4341, 42syl6bb 185 . . . . . . . . . . . . . . . 16 ((g Q 𝐶 P) → ( (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡 (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
4443ancoms 255 . . . . . . . . . . . . . . 15 ((𝐶 P g Q) → ( (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡 (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
456, 44sylan2 270 . . . . . . . . . . . . . 14 ((𝐶 P (B P g (2ndB))) → ( (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡 (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
4645anassrs 380 . . . . . . . . . . . . 13 (((𝐶 P B P) g (2ndB)) → ( (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡 (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
4746rexbidva 2317 . . . . . . . . . . . 12 ((𝐶 P B P) → (g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺) ↔ g (2ndB)𝑡 (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
4847ancoms 255 . . . . . . . . . . 11 ((B P 𝐶 P) → (g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺) ↔ g (2ndB)𝑡 (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
49 rexcom4 2571 . . . . . . . . . . 11 (g (2ndB)𝑡 (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)) ↔ 𝑡g (2ndB) (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺)))
5048, 49syl6bb 185 . . . . . . . . . 10 ((B P 𝐶 P) → (g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡g (2ndB) (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
5150adantr 261 . . . . . . . . 9 (((B P 𝐶 P) f Q) → (g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡g (2ndB) (2nd𝐶)(𝑡 = (g𝐺) x = ((f𝐺g)𝐺))))
52 df-rex 2306 . . . . . . . . . . 11 (𝑡 (2nd ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ 𝑡(𝑡 (2nd ‘(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, 32genpelvu 6496 . . . . . . . . . . . . 13 ((B P 𝐶 P) → (𝑡 (2nd ‘(B𝐹𝐶)) ↔ g (2ndB) (2nd𝐶)𝑡 = (g𝐺)))
5554anbi1d 438 . . . . . . . . . . . 12 ((B P 𝐶 P) → ((𝑡 (2nd ‘(B𝐹𝐶)) x = (f𝐺𝑡)) ↔ (g (2ndB) (2nd𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
5655exbidv 1703 . . . . . . . . . . 11 ((B P 𝐶 P) → (𝑡(𝑡 (2nd ‘(B𝐹𝐶)) x = (f𝐺𝑡)) ↔ 𝑡(g (2ndB) (2nd𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
5752, 56syl5bb 181 . . . . . . . . . 10 ((B P 𝐶 P) → (𝑡 (2nd ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ 𝑡(g (2ndB) (2nd𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
5857adantr 261 . . . . . . . . 9 (((B P 𝐶 P) f Q) → (𝑡 (2nd ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ 𝑡(g (2ndB) (2nd𝐶)𝑡 = (g𝐺) x = (f𝐺𝑡))))
5931, 51, 583bitr4rd 210 . . . . . . . 8 (((B P 𝐶 P) f Q) → (𝑡 (2nd ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺)))
603, 59sylan2 270 . . . . . . 7 (((B P 𝐶 P) (A P f (2ndA))) → (𝑡 (2nd ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺)))
6160anassrs 380 . . . . . 6 ((((B P 𝐶 P) A P) f (2ndA)) → (𝑡 (2nd ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺)))
6261rexbidva 2317 . . . . 5 (((B P 𝐶 P) A P) → (f (2ndA)𝑡 (2nd ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ f (2ndA)g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺)))
6362ancoms 255 . . . 4 ((A P (B P 𝐶 P)) → (f (2ndA)𝑡 (2nd ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ f (2ndA)g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺)))
64633impb 1099 . . 3 ((A P B P 𝐶 P) → (f (2ndA)𝑡 (2nd ‘(B𝐹𝐶))x = (f𝐺𝑡) ↔ f (2ndA)g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺)))
65 genpassg.5 . . . . . 6 ((f P g P) → (f𝐹g) P)
6665caovcl 5597 . . . . 5 ((B P 𝐶 P) → (B𝐹𝐶) P)
6753, 32genpelvu 6496 . . . . 5 ((A P (B𝐹𝐶) P) → (x (2nd ‘(A𝐹(B𝐹𝐶))) ↔ f (2ndA)𝑡 (2nd ‘(B𝐹𝐶))x = (f𝐺𝑡)))
6866, 67sylan2 270 . . . 4 ((A P (B P 𝐶 P)) → (x (2nd ‘(A𝐹(B𝐹𝐶))) ↔ f (2ndA)𝑡 (2nd ‘(B𝐹𝐶))x = (f𝐺𝑡)))
69683impb 1099 . . 3 ((A P B P 𝐶 P) → (x (2nd ‘(A𝐹(B𝐹𝐶))) ↔ f (2ndA)𝑡 (2nd ‘(B𝐹𝐶))x = (f𝐺𝑡)))
70 df-rex 2306 . . . . 5 (𝑡 (2nd ‘(A𝐹B)) (2nd𝐶)x = (𝑡𝐺) ↔ 𝑡(𝑡 (2nd ‘(A𝐹B)) (2nd𝐶)x = (𝑡𝐺)))
7153, 32genpelvu 6496 . . . . . . . 8 ((A P B P) → (𝑡 (2nd ‘(A𝐹B)) ↔ f (2ndA)g (2ndB)𝑡 = (f𝐺g)))
72713adant3 923 . . . . . . 7 ((A P B P 𝐶 P) → (𝑡 (2nd ‘(A𝐹B)) ↔ f (2ndA)g (2ndB)𝑡 = (f𝐺g)))
7372anbi1d 438 . . . . . 6 ((A P B P 𝐶 P) → ((𝑡 (2nd ‘(A𝐹B)) (2nd𝐶)x = (𝑡𝐺)) ↔ (f (2ndA)g (2ndB)𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺))))
7473exbidv 1703 . . . . 5 ((A P B P 𝐶 P) → (𝑡(𝑡 (2nd ‘(A𝐹B)) (2nd𝐶)x = (𝑡𝐺)) ↔ 𝑡(f (2ndA)g (2ndB)𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺))))
7570, 74syl5bb 181 . . . 4 ((A P B P 𝐶 P) → (𝑡 (2nd ‘(A𝐹B)) (2nd𝐶)x = (𝑡𝐺) ↔ 𝑡(f (2ndA)g (2ndB)𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺))))
7665caovcl 5597 . . . . . 6 ((A P B P) → (A𝐹B) P)
7753, 32genpelvu 6496 . . . . . 6 (((A𝐹B) P 𝐶 P) → (x (2nd ‘((A𝐹B)𝐹𝐶)) ↔ 𝑡 (2nd ‘(A𝐹B)) (2nd𝐶)x = (𝑡𝐺)))
7876, 77sylan 267 . . . . 5 (((A P B P) 𝐶 P) → (x (2nd ‘((A𝐹B)𝐹𝐶)) ↔ 𝑡 (2nd ‘(A𝐹B)) (2nd𝐶)x = (𝑡𝐺)))
79783impa 1098 . . . 4 ((A P B P 𝐶 P) → (x (2nd ‘((A𝐹B)𝐹𝐶)) ↔ 𝑡 (2nd ‘(A𝐹B)) (2nd𝐶)x = (𝑡𝐺)))
8032caovcl 5597 . . . . . . . . . . . . . . . . . . 19 ((f Q g Q) → (f𝐺g) Q)
81 elisset 2562 . . . . . . . . . . . . . . . . . . 19 ((f𝐺g) Q𝑡 𝑡 = (f𝐺g))
8280, 81syl 14 . . . . . . . . . . . . . . . . . 18 ((f Q g Q) → 𝑡 𝑡 = (f𝐺g))
8382biantrurd 289 . . . . . . . . . . . . . . . . 17 ((f Q g Q) → ( (2nd𝐶)x = ((f𝐺g)𝐺) ↔ (𝑡 𝑡 = (f𝐺g) (2nd𝐶)x = ((f𝐺g)𝐺))))
84 oveq1 5462 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = (f𝐺g) → (𝑡𝐺) = ((f𝐺g)𝐺))
8584eqeq2d 2048 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = (f𝐺g) → (x = (𝑡𝐺) ↔ x = ((f𝐺g)𝐺)))
8685rexbidv 2321 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = (f𝐺g) → ( (2nd𝐶)x = (𝑡𝐺) ↔ (2nd𝐶)x = ((f𝐺g)𝐺)))
8786pm5.32i 427 . . . . . . . . . . . . . . . . . . 19 ((𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)) ↔ (𝑡 = (f𝐺g) (2nd𝐶)x = ((f𝐺g)𝐺)))
8887exbii 1493 . . . . . . . . . . . . . . . . . 18 (𝑡(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)) ↔ 𝑡(𝑡 = (f𝐺g) (2nd𝐶)x = ((f𝐺g)𝐺)))
89 19.41v 1779 . . . . . . . . . . . . . . . . . 18 (𝑡(𝑡 = (f𝐺g) (2nd𝐶)x = ((f𝐺g)𝐺)) ↔ (𝑡 𝑡 = (f𝐺g) (2nd𝐶)x = ((f𝐺g)𝐺)))
9088, 89bitri 173 . . . . . . . . . . . . . . . . 17 (𝑡(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)) ↔ (𝑡 𝑡 = (f𝐺g) (2nd𝐶)x = ((f𝐺g)𝐺)))
9183, 90syl6bbr 187 . . . . . . . . . . . . . . . 16 ((f Q g Q) → ( (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺))))
926, 91sylan2 270 . . . . . . . . . . . . . . 15 ((f Q (B P g (2ndB))) → ( (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺))))
9392anassrs 380 . . . . . . . . . . . . . 14 (((f Q B P) g (2ndB)) → ( (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺))))
9493rexbidva 2317 . . . . . . . . . . . . 13 ((f Q B P) → (g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺) ↔ g (2ndB)𝑡(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺))))
95 rexcom4 2571 . . . . . . . . . . . . 13 (g (2ndB)𝑡(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)) ↔ 𝑡g (2ndB)(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)))
9694, 95syl6bb 185 . . . . . . . . . . . 12 ((f Q B P) → (g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡g (2ndB)(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺))))
9796ancoms 255 . . . . . . . . . . 11 ((B P f Q) → (g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡g (2ndB)(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺))))
983, 97sylan2 270 . . . . . . . . . 10 ((B P (A P f (2ndA))) → (g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡g (2ndB)(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺))))
9998anassrs 380 . . . . . . . . 9 (((B P A P) f (2ndA)) → (g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡g (2ndB)(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺))))
10099rexbidva 2317 . . . . . . . 8 ((B P A P) → (f (2ndA)g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺) ↔ f (2ndA)𝑡g (2ndB)(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺))))
101 rexcom4 2571 . . . . . . . 8 (f (2ndA)𝑡g (2ndB)(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)) ↔ 𝑡f (2ndA)g (2ndB)(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)))
102100, 101syl6bb 185 . . . . . . 7 ((B P A P) → (f (2ndA)g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡f (2ndA)g (2ndB)(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺))))
103 r19.41v 2460 . . . . . . . . . 10 (g (2ndB)(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)) ↔ (g (2ndB)𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)))
104103rexbii 2325 . . . . . . . . 9 (f (2ndA)g (2ndB)(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)) ↔ f (2ndA)(g (2ndB)𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)))
105 r19.41v 2460 . . . . . . . . 9 (f (2ndA)(g (2ndB)𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)) ↔ (f (2ndA)g (2ndB)𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)))
106104, 105bitri 173 . . . . . . . 8 (f (2ndA)g (2ndB)(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)) ↔ (f (2ndA)g (2ndB)𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)))
107106exbii 1493 . . . . . . 7 (𝑡f (2ndA)g (2ndB)(𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)) ↔ 𝑡(f (2ndA)g (2ndB)𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺)))
108102, 107syl6bb 185 . . . . . 6 ((B P A P) → (f (2ndA)g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡(f (2ndA)g (2ndB)𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺))))
109108ancoms 255 . . . . 5 ((A P B P) → (f (2ndA)g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡(f (2ndA)g (2ndB)𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺))))
1101093adant3 923 . . . 4 ((A P B P 𝐶 P) → (f (2ndA)g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺) ↔ 𝑡(f (2ndA)g (2ndB)𝑡 = (f𝐺g) (2nd𝐶)x = (𝑡𝐺))))
11175, 79, 1103bitr4d 209 . . 3 ((A P B P 𝐶 P) → (x (2nd ‘((A𝐹B)𝐹𝐶)) ↔ f (2ndA)g (2ndB) (2nd𝐶)x = ((f𝐺g)𝐺)))
11264, 69, 1113bitr4rd 210 . 2 ((A P B P 𝐶 P) → (x (2nd ‘((A𝐹B)𝐹𝐶)) ↔ x (2nd ‘(A𝐹(B𝐹𝐶)))))
113112eqrdv 2035 1 ((A P B P 𝐶 P) → (2nd ‘((A𝐹B)𝐹𝐶)) = (2nd ‘(A𝐹(B𝐹𝐶))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242  wex 1378   wcel 1390  wrex 2301  {crab 2304  cop 3370   × cxp 4286  dom cdm 4288  cfv 4845  (class class class)co 5455  cmpt2 5457  1st c1st 5707  2nd c2nd 5708  Qcnq 6264  Pcnp 6275
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-qs 6048  df-ni 6288  df-nqqs 6332  df-inp 6449
This theorem is referenced by:  genpassg  6509
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