Theorem initoeu2lem1 17952

Description: Lemma 1 for initoeu2 17954. (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 4690	. . . 4 ⊢ (∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ↔ ∃𝑓(𝐴𝐻𝐷) = {𝑓})
2		initoeu2lem.x	. . . . . . . . . . . 12 ⊢ 𝑋 = (Base‘𝐶)
3		eqid 2729	. . . . . . . . . . . 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 17706	. . . . . . . . . . 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 7039	. . . . . . . . . 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 17714	. . . . . . . . . . . . . . . . 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 3943	. . . . . . . . . . . . . . 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 17622	. . . . . . . . . . . . . . . . 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 17622	. . . . . . . . . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾)(⟨𝐴, 𝐵⟩ ⚬ 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ V
41		elsng 4599	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐺(⟨𝐴, 𝐵⟩ ⚬ 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) ∈ V
46		elsng 4599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 17951	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2561  Vcvv 3444   ⊆ wss 3911  {csn 4585  ⟨cop 4591  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  Hom chom 17207  compcco 17208  Catccat 17601  Invcinv 17683  Isociso 17684  InitOcinito 17919

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-cat 17605  df-cid 17606  df-sect 17685  df-inv 17686  df-iso 17687

This theorem is referenced by:  initoeu2lem2  17953