Theorem initoeu2lem1 17983

Description: Lemma 1 for initoeu2 17985. (Contributed by AV, 9-Apr-2020.)

Hypotheses

Ref	Expression
initoeu1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
initoeu1.a	⊢ (𝜑 → 𝐴 ∈ (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 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) ∧ (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

Step	Hyp	Ref	Expression
1		eusn 4697	. . . 4 ⊢ (∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ↔ ∃𝑓(𝐴𝐻𝐷) = {𝑓})
2		initoeu2lem.x	. . . . . . . . . . . 12 ⊢ 𝑋 = (Base‘𝐶)
3		eqid 2730	. . . . . . . . . . . 12 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
4		initoeu1.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ Cat)
5	4	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐶 ∈ Cat)
6		simpr2 1196	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
7	6	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐵 ∈ 𝑋)
8		simpr1 1195	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
9	8	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐴 ∈ 𝑋)
10		initoeu2lem.i	. . . . . . . . . . . 12 ⊢ 𝐼 = (Iso‘𝐶)
11	2, 3, 5, 7, 9, 10	invf 17737	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → (𝐵(Inv‘𝐶)𝐴):(𝐵𝐼𝐴)⟶(𝐴𝐼𝐵))
12		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐾 ∈ (𝐵𝐼𝐴))
13	11, 12	ffvelcdmd 7060	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵))
14		initoeu2lem.h	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐻 = (Hom ‘𝐶)
15	4	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → 𝐶 ∈ Cat)
16	2, 14, 10, 15, 8, 6	isohom 17745	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝐼𝐵) ⊆ (𝐴𝐻𝐵))
17	16	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → (𝐴𝐼𝐵) ⊆ (𝐴𝐻𝐵))
18	17	sselda 3949	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵))
19		initoeu2lem.o	. . . . . . . . . . . . . . . . . 18 ⊢ ⚬ = (comp‘𝐶)
20	15	ad4antr 732	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐶 ∈ Cat)
21	8	ad4antr 732	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐴 ∈ 𝑋)
22	6	ad4antr 732	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐵 ∈ 𝑋)
23		simpr3 1197	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → 𝐷 ∈ 𝑋)
24	23	ad4antr 732	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐷 ∈ 𝑋)
25		simplr 768	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵))
26		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐺 ∈ (𝐵𝐻𝐷))
27	2, 14, 19, 20, 21, 22, 24, 25, 26	catcocl 17653	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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))
28	15	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐶 ∈ Cat)
29	8	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐴 ∈ 𝑋)
30	6	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐵 ∈ 𝑋)
31	23	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐷 ∈ 𝑋)
32		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵))
33		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷))
34	2, 14, 19, 28, 29, 30, 31, 32, 33	catcocl 17653	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ ((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))
35	34	exp31 419	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (((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))))
36	35	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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))))
37	36	imp 406	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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 2818	. . . . . . . . . . . . . . . . . . . . . . . . . . 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}))
39	38	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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 7423	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾)(⟨𝐴, 𝐵⟩ ⚬ 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ V
41		elsng 4606	. . . . . . . . . . . . . . . . . . . . . . . . . . 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))
42	40, 41	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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))
43	39, 42	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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 2818	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7423	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐺(⟨𝐴, 𝐵⟩ ⚬ 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ V
46		elsng 4606	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
47	45, 46	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐴𝐻𝐷) = {𝑓} → ((𝐺(⟨𝐴, 𝐵⟩ ⚬ 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ {𝑓} ↔ (𝐺(⟨𝐴, 𝐵⟩ ⚬ 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓))
48	44, 47	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴𝐻𝐷) = {𝑓} → ((𝐺(⟨𝐴, 𝐵⟩ ⚬ 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ (𝐴𝐻𝐷) ↔ (𝐺(⟨𝐴, 𝐵⟩ ⚬ 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = 𝑓))
49	48	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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 2742	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
51	50	eqcoms 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
52	51	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)))
54		simp-4r 783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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 ⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))
59	4, 58, 2, 14, 10, 19	initoeu2lem0 17982	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (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))
60	53, 54, 55, 56, 57, 59	syl131anc 1385	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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))
61	60	exp43 436	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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)))))
62	61	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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)))))
63	52, 62	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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)))))
64	63	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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))))))
65	64	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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))))))
66	49, 65	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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))))))
67	66	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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))))))
68	43, 67	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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))))))
69	68	com23 86	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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))))))
70	69	ex 412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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)))))))
71	70	com24 95	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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)))))))
72	71	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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)))))))
73	37, 72	syld 47	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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)))))))
74	73	com25 99	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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)))))))
75	74	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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))))))
76	27, 75	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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)))))
77	76	ex 412	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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))))))
78	18, 77	mpdan 687	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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))))))
79	78	com15 101	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐴𝐻𝐷) → (𝐺 ∈ (𝐵𝐻𝐷) → ((𝐴𝐻𝐷) = {𝑓} → ((𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → 𝐺 = (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾))))))
80	79	imp 406	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → ((𝐴𝐻𝐷) = {𝑓} → ((𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → 𝐺 = (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾)))))
81	80	impcom 407	. . . . . . . . . . 11 ⊢ (((𝐴𝐻𝐷) = {𝑓} ∧ (𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷))) → ((𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐼𝐵)) → 𝐺 = (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾))))
82	81	com13 88	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 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))))
83	13, 82	mpdan 687	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → ((𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → (((𝐴𝐻𝐷) = {𝑓} ∧ (𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾))))
84	83	expimpd 453	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → (((𝐴𝐻𝐷) = {𝑓} ∧ (𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾))))
85	84	3impia 1117	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → (((𝐴𝐻𝐷) = {𝑓} ∧ (𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾)))
86	85	com12 32	. . . . . 6 ⊢ (((𝐴𝐻𝐷) = {𝑓} ∧ (𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷))) → ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾)))
87	86	ex 412	. . . . 5 ⊢ ((𝐴𝐻𝐷) = {𝑓} → ((𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾))))
88	87	exlimiv 1930	. . . 4 ⊢ (∃𝑓(𝐴𝐻𝐷) = {𝑓} → ((𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾))))
89	1, 88	sylbi 217	. . 3 ⊢ (∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) → ((𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾))))
90	89	3impib 1116	. 2 ⊢ ((∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → 𝐺 = (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾)))
91	90	com12 32	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → ((∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐺 = (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃!weu 2562  Vcvv 3450   ⊆ wss 3917  {csn 4592  ⟨cop 4598  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  Hom chom 17238  compcco 17239  Catccat 17632  Invcinv 17714  Isociso 17715  InitOcinito 17950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-cat 17636  df-cid 17637  df-sect 17716  df-inv 17717  df-iso 17718

This theorem is referenced by:  initoeu2lem2  17984