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Theorem initoeu2lem1 17340
 Description: Lemma 1 for initoeu2 17342. (Contributed by AV, 9-Apr-2020.)
Hypotheses
Ref Expression
initoeu1.c (𝜑𝐶 ∈ Cat)
initoeu1.a (𝜑𝐴 ∈ (InitO‘𝐶))
initoeu2lem.x 𝑋 = (Base‘𝐶)
initoeu2lem.h 𝐻 = (Hom ‘𝐶)
initoeu2lem.i 𝐼 = (Iso‘𝐶)
initoeu2lem.o = (comp‘𝐶)
Assertion
Ref Expression
initoeu2lem1 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → ((∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝐶,𝑓   𝜑,𝑓   𝐷,𝑓   𝑓,𝐹   𝑓,𝐺   𝑓,𝐼   𝑓,𝐾   𝑓,𝐻   𝑓,𝑋   ,𝑓

Proof of Theorem initoeu2lem1
StepHypRef Expression
1 eusn 4623 . . . 4 (∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ↔ ∃𝑓(𝐴𝐻𝐷) = {𝑓})
2 initoeu2lem.x . . . . . . . . . . . 12 𝑋 = (Base‘𝐶)
3 eqid 2758 . . . . . . . . . . . 12 (Inv‘𝐶) = (Inv‘𝐶)
4 initoeu1.c . . . . . . . . . . . . 13 (𝜑𝐶 ∈ Cat)
54ad2antrr 725 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐶 ∈ Cat)
6 simpr2 1192 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) → 𝐵𝑋)
76adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐵𝑋)
8 simpr1 1191 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) → 𝐴𝑋)
98adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐴𝑋)
10 initoeu2lem.i . . . . . . . . . . . 12 𝐼 = (Iso‘𝐶)
112, 3, 5, 7, 9, 10invf 17097 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → (𝐵(Inv‘𝐶)𝐴):(𝐵𝐼𝐴)⟶(𝐴𝐼𝐵))
12 simpr 488 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐾 ∈ (𝐵𝐼𝐴))
1311, 12ffvelrnd 6843 . . . . . . . . . 10 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵))
14 initoeu2lem.h . . . . . . . . . . . . . . . . . 18 𝐻 = (Hom ‘𝐶)
154adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) → 𝐶 ∈ Cat)
162, 14, 10, 15, 8, 6isohom 17105 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) → (𝐴𝐼𝐵) ⊆ (𝐴𝐻𝐵))
1716adantr 484 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → (𝐴𝐼𝐵) ⊆ (𝐴𝐻𝐵))
1817sselda 3892 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵))
19 initoeu2lem.o . . . . . . . . . . . . . . . . . 18 = (comp‘𝐶)
2015ad4antr 731 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐶 ∈ Cat)
218ad4antr 731 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐴𝑋)
226ad4antr 731 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐵𝑋)
23 simpr3 1193 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) → 𝐷𝑋)
2423ad4antr 731 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐷𝑋)
25 simplr 768 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵))
26 simpr 488 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐺 ∈ (𝐵𝐻𝐷))
272, 14, 19, 20, 21, 22, 24, 25, 26catcocl 17014 . . . . . . . . . . . . . . . . 17 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷))
2815ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐶 ∈ Cat)
298ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐴𝑋)
306ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐵𝑋)
3123ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐷𝑋)
32 simplr 768 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵))
33 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))
342, 14, 19, 28, 29, 30, 31, 32, 33catcocl 17014 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷))
3534exp31 423 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) → (((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵) → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷))))
3635ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → (((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵) → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷))))
3736imp 410 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷)))
38 eleq2 2840 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴𝐻𝐷) = {𝑓} → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) ↔ ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ {𝑓}))
3938adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ (𝐴𝐻𝐷) = {𝑓}) → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) ↔ ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ {𝑓}))
40 ovex 7183 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ V
41 elsng 4536 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ V → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ {𝑓} ↔ ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓))
4240, 41mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ (𝐴𝐻𝐷) = {𝑓}) → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ {𝑓} ↔ ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓))
4339, 42bitrd 282 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ (𝐴𝐻𝐷) = {𝑓}) → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) ↔ ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓))
44 eleq2 2840 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐴𝐻𝐷) = {𝑓} → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) ↔ (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ {𝑓}))
45 ovex 7183 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ V
46 elsng 4536 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ V → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ {𝑓} ↔ (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓))
4745, 46mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐴𝐻𝐷) = {𝑓} → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ {𝑓} ↔ (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓))
4844, 47bitrd 282 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴𝐻𝐷) = {𝑓} → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) ↔ (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓))
4948adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ (𝐴𝐻𝐷) = {𝑓}) → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) ↔ (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓))
50 eqeq2 2770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 = (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓 ↔ ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾))))
5150eqcoms 2766 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓 → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓 ↔ ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾))))
5251adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓) → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓 ↔ ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾))))
53 simp-4l 782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾))) ∧ (𝐺 ∈ (𝐵𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷))) → (𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)))
54 simp-4r 783 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾))) ∧ (𝐺 ∈ (𝐵𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷))) → 𝐾 ∈ (𝐵𝐼𝐴))
55 simprr 772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾))) ∧ (𝐺 ∈ (𝐵𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷))) → 𝐹 ∈ (𝐴𝐻𝐷))
56 simprl 770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾))) ∧ (𝐺 ∈ (𝐵𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷))) → 𝐺 ∈ (𝐵𝐻𝐷))
57 simplr 768 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾))) ∧ (𝐺 ∈ (𝐵𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷))) → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)))
58 initoeu1.a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑𝐴 ∈ (InitO‘𝐶))
594, 58, 2, 14, 10, 19initoeu2lem0 17339 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) ∧ ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾))) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))
6053, 54, 55, 56, 57, 59syl131anc 1380 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾))) ∧ (𝐺 ∈ (𝐵𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))
6160exp43 440 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) → (𝐺 ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))))
6261adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓) → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) → (𝐺 ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))))
6352, 62sylbid 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓) → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓 → (𝐺 ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))))
6463ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓 → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓 → (𝐺 ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))))
6564adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ (𝐴𝐻𝐷) = {𝑓}) → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓 → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓 → (𝐺 ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))))
6649, 65sylbid 243 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ (𝐴𝐻𝐷) = {𝑓}) → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓 → (𝐺 ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))))
6766com23 86 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ (𝐴𝐻𝐷) = {𝑓}) → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓 → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) → (𝐺 ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))))
6843, 67sylbid 243 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ (𝐴𝐻𝐷) = {𝑓}) → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) → (𝐺 ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))))
6968com23 86 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ (𝐴𝐻𝐷) = {𝑓}) → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) → (𝐺 ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))))
7069ex 416 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → ((𝐴𝐻𝐷) = {𝑓} → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) → (𝐺 ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))))))
7170com24 95 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) → ((𝐴𝐻𝐷) = {𝑓} → (𝐺 ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))))))
7271adantr 484 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) → (((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) → ((𝐴𝐻𝐷) = {𝑓} → (𝐺 ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))))))
7337, 72syld 47 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) → ((𝐴𝐻𝐷) = {𝑓} → (𝐺 ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))))))
7473com25 99 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) → (𝐺 ∈ (𝐵𝐻𝐷) → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) → ((𝐴𝐻𝐷) = {𝑓} → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))))))
7574imp 410 . . . . . . . . . . . . . . . . 17 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) → ((𝐴𝐻𝐷) = {𝑓} → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))))
7627, 75mpd 15 . . . . . . . . . . . . . . . 16 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → ((𝐴𝐻𝐷) = {𝑓} → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))))
7776ex 416 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) → (𝐺 ∈ (𝐵𝐻𝐷) → ((𝐴𝐻𝐷) = {𝑓} → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))))
7818, 77mpdan 686 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → (𝐺 ∈ (𝐵𝐻𝐷) → ((𝐴𝐻𝐷) = {𝑓} → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → (𝐹 ∈ (𝐴𝐻𝐷) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))))
7978com15 101 . . . . . . . . . . . . 13 (𝐹 ∈ (𝐴𝐻𝐷) → (𝐺 ∈ (𝐵𝐻𝐷) → ((𝐴𝐻𝐷) = {𝑓} → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))))
8079imp 410 . . . . . . . . . . . 12 ((𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → ((𝐴𝐻𝐷) = {𝑓} → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))))
8180impcom 411 . . . . . . . . . . 11 (((𝐴𝐻𝐷) = {𝑓} ∧ (𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷))) → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))
8281com13 88 . . . . . . . . . 10 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → (((𝐴𝐻𝐷) = {𝑓} ∧ (𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))
8313, 82mpdan 686 . . . . . . . . 9 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → (((𝐴𝐻𝐷) = {𝑓} ∧ (𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))
8483expimpd 457 . . . . . . . 8 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) → ((𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → (((𝐴𝐻𝐷) = {𝑓} ∧ (𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))
85843impia 1114 . . . . . . 7 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → (((𝐴𝐻𝐷) = {𝑓} ∧ (𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))
8685com12 32 . . . . . 6 (((𝐴𝐻𝐷) = {𝑓} ∧ (𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷))) → ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))
8786ex 416 . . . . 5 ((𝐴𝐻𝐷) = {𝑓} → ((𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))
8887exlimiv 1931 . . . 4 (∃𝑓(𝐴𝐻𝐷) = {𝑓} → ((𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))
891, 88sylbi 220 . . 3 (∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) → ((𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))
90893impib 1113 . 2 ((∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))
9190com12 32 1 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → ((∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃!weu 2587  Vcvv 3409   ⊆ wss 3858  {csn 4522  ⟨cop 4528  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  Hom chom 16634  compcco 16635  Catccat 16993  Invcinv 17074  Isociso 17075  InitOcinito 17307 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-cat 16997  df-cid 16998  df-sect 17076  df-inv 17077  df-iso 17078 This theorem is referenced by:  initoeu2lem2  17341
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