Theorem initoeu2lem1 18028

Description: Lemma 1 for initoeu2 18030. (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 4688	. . . 4 ⊢ (∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ↔ ∃𝑓(𝐴𝐻𝐷) = {𝑓})
2		initoeu2lem.x	. . . . . . . . . . . 12 ⊢ 𝑋 = (Base‘𝐶)
3		eqid 2761	. . . . . . . . . . . 12 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
4		initoeu1.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ Cat)
5	4	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐶 ∈ Cat)
6		simpr2 1208	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
7	6	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → 𝐵 ∈ 𝑋)
8		simpr1 1207	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
9	8	adantr 484	. . . . . . . . . . . 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 17782	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → (𝐵(Inv‘𝐶)𝐴):(𝐵𝐼𝐴)⟶(𝐴𝐼𝐵))
12		simpr 488	. . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → 𝐶 ∈ Cat)
16	2, 14, 10, 15, 8, 6	isohom 17790	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝐼𝐵) ⊆ (𝐴𝐻𝐵))
17	16	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ 𝐾 ∈ (𝐵𝐼𝐴)) → (𝐴𝐼𝐵) ⊆ (𝐴𝐻𝐵))
18	17	sselda 3936	. . . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . . . . . 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 1209	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → 𝐷 ∈ 𝑋)
24	23	ad4antr 742	. . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . 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 17698	. . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ ((𝐵(Inv‘𝐶)𝐴)‘𝐾) ∈ (𝐴𝐻𝐵)) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → 𝐷 ∈ 𝑋)
32		simplr 778	. . . . . . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . . . 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 17698	. . . . . . . . . . . . . . . . . . . . . . 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 423	. . . . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . 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 2850	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4595	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . . . . . . . . . . . . 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 2850	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 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))) → 𝐹 ∈ (𝐴𝐻𝐷))
56		simprl 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18027	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1401	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . 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 697	. . . . . . . . . . . . . 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 410	. . . . . . . . . . . 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 411	. . . . . . . . . . 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 697	. . . . . . . . 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 457	. . . . . . . 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 1129	. . . . . . 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 416	. . . . 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 1949	. . . 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 219	. . 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 1128	. 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 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∃!weu 2594  Vcvv 3453   ⊆ wss 3904  {csn 4581  ⟨cop 4587  ‘cfv 6515  (class class class)co 7390  Basecbs 17226  Hom chom 17278  compcco 17279  Catccat 17677  Invcinv 17759  Isociso 17760  InitOcinito 17995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7964  df-2nd 7965  df-cat 17681  df-cid 17682  df-sect 17761  df-inv 17762  df-iso 17763

This theorem is referenced by:  initoeu2lem2  18029