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Theorem initoeu2lem1 17262
Description: Lemma 1 for initoeu2 17264. (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 4658 . . . 4 (∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ↔ ∃𝑓(𝐴𝐻𝐷) = {𝑓})
2 initoeu2lem.x . . . . . . . . . . . 12 𝑋 = (Base‘𝐶)
3 eqid 2818 . . . . . . . . . . . 12 (Inv‘𝐶) = (Inv‘𝐶)
4 initoeu1.c . . . . . . . . . . . . 13 (𝜑𝐶 ∈ Cat)
54ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐶 ∈ Cat)
6 simpr2 1187 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) → 𝐵𝑋)
76adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐵𝑋)
8 simpr1 1186 . . . . . . . . . . . . 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 17026 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → (𝐵(Inv‘𝐶)𝐴):(𝐵𝐼𝐴)⟶(𝐴𝐼𝐵))
12 simpr 485 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐾 ∈ (𝐵𝐼𝐴))
1311, 12ffvelrnd 6844 . . . . . . . . . 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 17034 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) → (𝐴𝐼𝐵) ⊆ (𝐴𝐻𝐵))
1716adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → (𝐴𝐼𝐵) ⊆ (𝐴𝐻𝐵))
1817sselda 3964 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵))
19 initoeu2lem.o . . . . . . . . . . . . . . . . . 18 = (comp‘𝐶)
2015ad4antr 728 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐶 ∈ Cat)
218ad4antr 728 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐴𝑋)
226ad4antr 728 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐵𝑋)
23 simpr3 1188 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) → 𝐷𝑋)
2423ad4antr 728 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐷𝑋)
25 simplr 765 . . . . . . . . . . . . . . . . . 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 16944 . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐶 ∈ Cat)
298ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐴𝑋)
306ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐵𝑋)
3123ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐷𝑋)
32 simplr 765 . . . . . . . . . . . . . . . . . . . . . . . 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 16944 . . . . . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . . . . 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 2898 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7178 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ V
41 elsng 4571 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 280 . . . . . . . . . . . . . . . . . . . . . . . . 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 2898 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7178 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ V
46 elsng 4571 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 280 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 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))) → 𝐹 ∈ (𝐴𝐻𝐷))
56 simprl 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))) → 𝐺 ∈ (𝐵𝐻𝐷))
57 simplr 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 17261 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1375 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 241 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 241 . . . . . . . . . . . . . . . . . . . . . . . . . 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 241 . . . . . . . . . . . . . . . . . . . . . . . 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 683 . . . . . . . . . . . . . 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 683 . . . . . . . . 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 1109 . . . . . . 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 1922 . . . 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 218 . . 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 1108 . 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 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  ∃!weu 2646  Vcvv 3492  wss 3933  {csn 4557  cop 4563  cfv 6348  (class class class)co 7145  Basecbs 16471  Hom chom 16564  compcco 16565  Catccat 16923  Invcinv 17003  Isociso 17004  InitOcinito 17236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-cat 16927  df-cid 16928  df-sect 17005  df-inv 17006  df-iso 17007
This theorem is referenced by:  initoeu2lem2  17263
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