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Theorem initoeu2lem1 17900
Description: Lemma 1 for initoeu2 17902. (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 4691 . . . 4 (∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ↔ ∃𝑓(𝐴𝐻𝐷) = {𝑓})
2 initoeu2lem.x . . . . . . . . . . . 12 𝑋 = (Base‘𝐶)
3 eqid 2736 . . . . . . . . . . . 12 (Inv‘𝐶) = (Inv‘𝐶)
4 initoeu1.c . . . . . . . . . . . . 13 (𝜑𝐶 ∈ Cat)
54ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐶 ∈ Cat)
6 simpr2 1195 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) → 𝐵𝑋)
76adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐵𝑋)
8 simpr1 1194 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) → 𝐴𝑋)
98adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐴𝑋)
10 initoeu2lem.i . . . . . . . . . . . 12 𝐼 = (Iso‘𝐶)
112, 3, 5, 7, 9, 10invf 17651 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → (𝐵(Inv‘𝐶)𝐴):(𝐵𝐼𝐴)⟶(𝐴𝐼𝐵))
12 simpr 485 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐾 ∈ (𝐵𝐼𝐴))
1311, 12ffvelcdmd 7036 . . . . . . . . . 10 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵))
14 initoeu2lem.h . . . . . . . . . . . . . . . . . 18 𝐻 = (Hom ‘𝐶)
154adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) → 𝐶 ∈ Cat)
162, 14, 10, 15, 8, 6isohom 17659 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) → (𝐴𝐼𝐵) ⊆ (𝐴𝐻𝐵))
1716adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → (𝐴𝐼𝐵) ⊆ (𝐴𝐻𝐵))
1817sselda 3944 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵))
19 initoeu2lem.o . . . . . . . . . . . . . . . . . 18 = (comp‘𝐶)
2015ad4antr 730 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐶 ∈ Cat)
218ad4antr 730 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐴𝑋)
226ad4antr 730 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐵𝑋)
23 simpr3 1196 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) → 𝐷𝑋)
2423ad4antr 730 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐷𝑋)
25 simplr 767 . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . 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 17565 . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐶 ∈ Cat)
298ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐴𝑋)
306ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐵𝑋)
3123ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐷𝑋)
32 simplr 767 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵))
33 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . 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 17565 . . . . . . . . . . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . . 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 7390 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ V
41 elsng 4600 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 278 . . . . . . . . . . . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7390 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ V
46 elsng 4600 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴𝐻𝐷) = {𝑓} → ((𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) ↔ (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓))
4948adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 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))) → 𝐾 ∈ (𝐵𝐼𝐴))
55 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 17899 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1383 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 437 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 239 . . . . . . . . . . . . . . . . . . . . . . . . . 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 239 . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . 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 685 . . . . . . . . . . . . . 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 407 . . . . . . . . . . . 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 408 . . . . . . . . . . 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 685 . . . . . . . . 9 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → (((𝐴𝐻𝐷) = {𝑓} ∧ (𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))
8483expimpd 454 . . . . . . . 8 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) → ((𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → (((𝐴𝐻𝐷) = {𝑓} ∧ (𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))
85843impia 1117 . . . . . . 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 413 . . . . 5 ((𝐴𝐻𝐷) = {𝑓} → ((𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))))
8887exlimiv 1933 . . . 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 216 . . 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 1116 . 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 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  ∃!weu 2566  Vcvv 3445  wss 3910  {csn 4586  cop 4592  cfv 6496  (class class class)co 7357  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544  Invcinv 17628  Isociso 17629  InitOcinito 17867
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-cat 17548  df-cid 17549  df-sect 17630  df-inv 17631  df-iso 17632
This theorem is referenced by:  initoeu2lem2  17901
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