Theorem iscatd2 16342

Description: Version of iscatd 16334 with a uniform assumption list, for increased proof sharing capabilities. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
iscatd2.b	⊢ (𝜑 → 𝐵 = (Base‘𝐶))
iscatd2.h	⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
iscatd2.o	⊢ (𝜑 → · = (comp‘𝐶))
iscatd2.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
iscatd2.ps	⊢ (𝜓 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
iscatd2.1	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1 ∈ (𝑦𝐻𝑦))
iscatd2.2	⊢ ((𝜑 ∧ 𝜓) → ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓)
iscatd2.3	⊢ ((𝜑 ∧ 𝜓) → (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑔)
iscatd2.4	⊢ ((𝜑 ∧ 𝜓) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
iscatd2.5	⊢ ((𝜑 ∧ 𝜓) → ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))

Assertion

Ref	Expression
iscatd2	⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 1 )))

Distinct variable groups:   𝑓,𝑔,𝑘,𝑤,𝑥,𝑧, 1   𝑦,𝑓,𝐵,𝑔,𝑘,𝑤,𝑥,𝑧   𝐶,𝑔,𝑘,𝑤,𝑦,𝑧   𝑓,𝐻,𝑔,𝑘,𝑤,𝑥,𝑦,𝑧   𝜑,𝑓,𝑔,𝑘,𝑤,𝑥,𝑦,𝑧   · ,𝑓,𝑔,𝑘,𝑤,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔,𝑘)   𝐶(𝑥,𝑓)   1 (𝑦)   𝑉(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔,𝑘)

Proof of Theorem iscatd2

Dummy variables 𝑎 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscatd2.b	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
2		iscatd2.h	. . 3 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
3		iscatd2.o	. . 3 ⊢ (𝜑 → · = (comp‘𝐶))
4		iscatd2.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
5		iscatd2.1	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1 ∈ (𝑦𝐻𝑦))
6		ne0i 3921	. . . . . . 7 ⊢ ( 1 ∈ (𝑦𝐻𝑦) → (𝑦𝐻𝑦) ≠ ∅)
7	5, 6	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦𝐻𝑦) ≠ ∅)
8	7	3ad2antr1 1226	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑎𝐻𝑦))) → (𝑦𝐻𝑦) ≠ ∅)
9		n0 3931	. . . . 5 ⊢ ((𝑦𝐻𝑦) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝑦𝐻𝑦))
10	8, 9	sylib 208	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑎𝐻𝑦))) → ∃𝑔 𝑔 ∈ (𝑦𝐻𝑦))
11		n0 3931	. . . . 5 ⊢ ((𝑦𝐻𝑦) ≠ ∅ ↔ ∃𝑘 𝑘 ∈ (𝑦𝐻𝑦))
12	8, 11	sylib 208	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑎𝐻𝑦))) → ∃𝑘 𝑘 ∈ (𝑦𝐻𝑦))
13		eeanv 2182	. . . . 5 ⊢ (∃𝑔∃𝑘(𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)) ↔ (∃𝑔 𝑔 ∈ (𝑦𝐻𝑦) ∧ ∃𝑘 𝑘 ∈ (𝑦𝐻𝑦)))
14		simpll 790	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝜑)
15		simplr2 1104	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝑎 ∈ 𝐵)
16		simplr1 1103	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝑦 ∈ 𝐵)
17	15, 16	jca 554	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → (𝑎 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
18		simplr3 1105	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝑟 ∈ (𝑎𝐻𝑦))
19		simprl 794	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝑔 ∈ (𝑦𝐻𝑦))
20		simprr 796	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝑘 ∈ (𝑦𝐻𝑦))
21	18, 19, 20	3jca 1242	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))
22		iscatd2.ps	. . . . . . . . . . . . . . 15 ⊢ (𝜓 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
23		simplll 798	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → 𝑥 = 𝑎)
24	23	eleq1d 2686	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑥 ∈ 𝐵 ↔ 𝑎 ∈ 𝐵))
25	24	anbi1d 741	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (𝑎 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
26		simpllr 799	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → 𝑧 = 𝑦)
27	26	eleq1d 2686	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑧 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
28		simplr 792	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → 𝑤 = 𝑦)
29	28	eleq1d 2686	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑤 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
30	27, 29	anbi12d 747	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
31		anidm 676	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ 𝑦 ∈ 𝐵)
32	30, 31	syl6bb 276	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ↔ 𝑦 ∈ 𝐵))
33		simpr 477	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → 𝑓 = 𝑟)
34	23	oveq1d 6665	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑥𝐻𝑦) = (𝑎𝐻𝑦))
35	33, 34	eleq12d 2695	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑟 ∈ (𝑎𝐻𝑦)))
36	26	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑦𝐻𝑧) = (𝑦𝐻𝑦))
37	36	eleq2d 2687	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑔 ∈ (𝑦𝐻𝑧) ↔ 𝑔 ∈ (𝑦𝐻𝑦)))
38	26, 28	oveq12d 6668	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑧𝐻𝑤) = (𝑦𝐻𝑦))
39	38	eleq2d 2687	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑘 ∈ (𝑧𝐻𝑤) ↔ 𝑘 ∈ (𝑦𝐻𝑦)))
40	35, 37, 39	3anbi123d 1399	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))))
41	25, 32, 40	3anbi123d 1399	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ ((𝑎 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))))
42	22, 41	syl5bb 272	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝜓 ↔ ((𝑎 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))))
43	42	anbi2d 740	. . . . . . . . . . . . 13 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ((𝑎 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))))))
44	23	opeq1d 4408	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑦⟩)
45	44	oveq1d 6665	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (⟨𝑥, 𝑦⟩ · 𝑦) = (⟨𝑎, 𝑦⟩ · 𝑦))
46		eqidd 2623	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → 1 = 1 )
47	45, 46, 33	oveq123d 6671	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = ( 1 (⟨𝑎, 𝑦⟩ · 𝑦)𝑟))
48	47, 33	eqeq12d 2637	. . . . . . . . . . . . 13 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓 ↔ ( 1 (⟨𝑎, 𝑦⟩ · 𝑦)𝑟) = 𝑟))
49	43, 48	imbi12d 334	. . . . . . . . . . . 12 ⊢ ((((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (((𝜑 ∧ 𝜓) → ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓) ↔ ((𝜑 ∧ ((𝑎 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))) → ( 1 (⟨𝑎, 𝑦⟩ · 𝑦)𝑟) = 𝑟)))
50	49	sbiedv 2410	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) ∧ 𝑤 = 𝑦) → ([𝑟 / 𝑓]((𝜑 ∧ 𝜓) → ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓) ↔ ((𝜑 ∧ ((𝑎 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))) → ( 1 (⟨𝑎, 𝑦⟩ · 𝑦)𝑟) = 𝑟)))
51	50	sbiedv 2410	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑧 = 𝑦) → ([𝑦 / 𝑤][𝑟 / 𝑓]((𝜑 ∧ 𝜓) → ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓) ↔ ((𝜑 ∧ ((𝑎 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))) → ( 1 (⟨𝑎, 𝑦⟩ · 𝑦)𝑟) = 𝑟)))
52	51	sbiedv 2410	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ([𝑦 / 𝑧][𝑦 / 𝑤][𝑟 / 𝑓]((𝜑 ∧ 𝜓) → ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓) ↔ ((𝜑 ∧ ((𝑎 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))) → ( 1 (⟨𝑎, 𝑦⟩ · 𝑦)𝑟) = 𝑟)))
53		iscatd2.2	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓)
54	53	sbt 2419	. . . . . . . . . . 11 ⊢ [𝑟 / 𝑓]((𝜑 ∧ 𝜓) → ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓)
55	54	sbt 2419	. . . . . . . . . 10 ⊢ [𝑦 / 𝑤][𝑟 / 𝑓]((𝜑 ∧ 𝜓) → ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓)
56	55	sbt 2419	. . . . . . . . 9 ⊢ [𝑦 / 𝑧][𝑦 / 𝑤][𝑟 / 𝑓]((𝜑 ∧ 𝜓) → ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓)
57	52, 56	chvarv 2263	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))) → ( 1 (⟨𝑎, 𝑦⟩ · 𝑦)𝑟) = 𝑟)
58	14, 17, 16, 21, 57	syl13anc 1328	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → ( 1 (⟨𝑎, 𝑦⟩ · 𝑦)𝑟) = 𝑟)
59	58	ex 450	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑎𝐻𝑦))) → ((𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)) → ( 1 (⟨𝑎, 𝑦⟩ · 𝑦)𝑟) = 𝑟))
60	59	exlimdvv 1862	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑎𝐻𝑦))) → (∃𝑔∃𝑘(𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)) → ( 1 (⟨𝑎, 𝑦⟩ · 𝑦)𝑟) = 𝑟))
61	13, 60	syl5bir 233	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑎𝐻𝑦))) → ((∃𝑔 𝑔 ∈ (𝑦𝐻𝑦) ∧ ∃𝑘 𝑘 ∈ (𝑦𝐻𝑦)) → ( 1 (⟨𝑎, 𝑦⟩ · 𝑦)𝑟) = 𝑟))
62	10, 12, 61	mp2and 715	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑎𝐻𝑦))) → ( 1 (⟨𝑎, 𝑦⟩ · 𝑦)𝑟) = 𝑟)
63	7	3ad2antr1 1226	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) → (𝑦𝐻𝑦) ≠ ∅)
64		n0 3931	. . . . 5 ⊢ ((𝑦𝐻𝑦) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑦𝐻𝑦))
65	63, 64	sylib 208	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) → ∃𝑓 𝑓 ∈ (𝑦𝐻𝑦))
66		simpr2 1068	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) → 𝑎 ∈ 𝐵)
67	7	ralrimiva 2966	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝑦𝐻𝑦) ≠ ∅)
68	67	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) → ∀𝑦 ∈ 𝐵 (𝑦𝐻𝑦) ≠ ∅)
69		id 22	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → 𝑦 = 𝑎)
70	69, 69	oveq12d 6668	. . . . . . . 8 ⊢ (𝑦 = 𝑎 → (𝑦𝐻𝑦) = (𝑎𝐻𝑎))
71	70	neeq1d 2853	. . . . . . 7 ⊢ (𝑦 = 𝑎 → ((𝑦𝐻𝑦) ≠ ∅ ↔ (𝑎𝐻𝑎) ≠ ∅))
72	71	rspcv 3305	. . . . . 6 ⊢ (𝑎 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 (𝑦𝐻𝑦) ≠ ∅ → (𝑎𝐻𝑎) ≠ ∅))
73	66, 68, 72	sylc 65	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) → (𝑎𝐻𝑎) ≠ ∅)
74		n0 3931	. . . . 5 ⊢ ((𝑎𝐻𝑎) ≠ ∅ ↔ ∃𝑘 𝑘 ∈ (𝑎𝐻𝑎))
75	73, 74	sylib 208	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) → ∃𝑘 𝑘 ∈ (𝑎𝐻𝑎))
76		eeanv 2182	. . . . 5 ⊢ (∃𝑓∃𝑘(𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎)) ↔ (∃𝑓 𝑓 ∈ (𝑦𝐻𝑦) ∧ ∃𝑘 𝑘 ∈ (𝑎𝐻𝑎)))
77		simpll 790	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝜑)
78		simplr1 1103	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝑦 ∈ 𝐵)
79		simplr2 1104	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝑎 ∈ 𝐵)
80		simprl 794	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝑓 ∈ (𝑦𝐻𝑦))
81		simplr3 1105	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝑟 ∈ (𝑦𝐻𝑎))
82		simprr 796	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝑘 ∈ (𝑎𝐻𝑎))
83	80, 81, 82	3jca 1242	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))
84		simplll 798	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → 𝑥 = 𝑦)
85	84	eleq1d 2686	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
86	85	anbi1d 741	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
87	86, 31	syl6bb 276	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ 𝑦 ∈ 𝐵))
88		simpllr 799	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → 𝑧 = 𝑎)
89	88	eleq1d 2686	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑧 ∈ 𝐵 ↔ 𝑎 ∈ 𝐵))
90		simplr 792	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → 𝑤 = 𝑎)
91	90	eleq1d 2686	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑤 ∈ 𝐵 ↔ 𝑎 ∈ 𝐵))
92	89, 91	anbi12d 747	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ↔ (𝑎 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵)))
93		anidm 676	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ↔ 𝑎 ∈ 𝐵)
94	92, 93	syl6bb 276	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ↔ 𝑎 ∈ 𝐵))
95	84	oveq1d 6665	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑥𝐻𝑦) = (𝑦𝐻𝑦))
96	95	eleq2d 2687	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑦𝐻𝑦)))
97		simpr 477	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → 𝑔 = 𝑟)
98	88	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑦𝐻𝑧) = (𝑦𝐻𝑎))
99	97, 98	eleq12d 2695	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑔 ∈ (𝑦𝐻𝑧) ↔ 𝑟 ∈ (𝑦𝐻𝑎)))
100	88, 90	oveq12d 6668	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑧𝐻𝑤) = (𝑎𝐻𝑎))
101	100	eleq2d 2687	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑘 ∈ (𝑧𝐻𝑤) ↔ 𝑘 ∈ (𝑎𝐻𝑎)))
102	96, 99, 101	3anbi123d 1399	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎))))
103	87, 94, 102	3anbi123d 1399	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))))
104	22, 103	syl5bb 272	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝜓 ↔ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))))
105	104	anbi2d 740	. . . . . . . . . . . . 13 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎))))))
106	88	oveq2d 6666	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (⟨𝑦, 𝑦⟩ · 𝑧) = (⟨𝑦, 𝑦⟩ · 𝑎))
107		eqidd 2623	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → 1 = 1 )
108	106, 97, 107	oveq123d 6671	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = (𝑟(⟨𝑦, 𝑦⟩ · 𝑎) 1 ))
109	108, 97	eqeq12d 2637	. . . . . . . . . . . . 13 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑔 ↔ (𝑟(⟨𝑦, 𝑦⟩ · 𝑎) 1 ) = 𝑟))
110	105, 109	imbi12d 334	. . . . . . . . . . . 12 ⊢ ((((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (((𝜑 ∧ 𝜓) → (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑔) ↔ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))) → (𝑟(⟨𝑦, 𝑦⟩ · 𝑎) 1 ) = 𝑟)))
111	110	sbiedv 2410	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) ∧ 𝑤 = 𝑎) → ([𝑟 / 𝑔]((𝜑 ∧ 𝜓) → (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑔) ↔ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))) → (𝑟(⟨𝑦, 𝑦⟩ · 𝑎) 1 ) = 𝑟)))
112	111	sbiedv 2410	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑎) → ([𝑎 / 𝑤][𝑟 / 𝑔]((𝜑 ∧ 𝜓) → (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑔) ↔ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))) → (𝑟(⟨𝑦, 𝑦⟩ · 𝑎) 1 ) = 𝑟)))
113	112	sbiedv 2410	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ([𝑎 / 𝑧][𝑎 / 𝑤][𝑟 / 𝑔]((𝜑 ∧ 𝜓) → (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑔) ↔ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))) → (𝑟(⟨𝑦, 𝑦⟩ · 𝑎) 1 ) = 𝑟)))
114		iscatd2.3	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑔)
115	114	sbt 2419	. . . . . . . . . . 11 ⊢ [𝑟 / 𝑔]((𝜑 ∧ 𝜓) → (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑔)
116	115	sbt 2419	. . . . . . . . . 10 ⊢ [𝑎 / 𝑤][𝑟 / 𝑔]((𝜑 ∧ 𝜓) → (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑔)
117	116	sbt 2419	. . . . . . . . 9 ⊢ [𝑎 / 𝑧][𝑎 / 𝑤][𝑟 / 𝑔]((𝜑 ∧ 𝜓) → (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑔)
118	113, 117	chvarv 2263	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))) → (𝑟(⟨𝑦, 𝑦⟩ · 𝑎) 1 ) = 𝑟)
119	77, 78, 79, 83, 118	syl13anc 1328	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → (𝑟(⟨𝑦, 𝑦⟩ · 𝑎) 1 ) = 𝑟)
120	119	ex 450	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) → ((𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎)) → (𝑟(⟨𝑦, 𝑦⟩ · 𝑎) 1 ) = 𝑟))
121	120	exlimdvv 1862	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) → (∃𝑓∃𝑘(𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎)) → (𝑟(⟨𝑦, 𝑦⟩ · 𝑎) 1 ) = 𝑟))
122	76, 121	syl5bir 233	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) → ((∃𝑓 𝑓 ∈ (𝑦𝐻𝑦) ∧ ∃𝑘 𝑘 ∈ (𝑎𝐻𝑎)) → (𝑟(⟨𝑦, 𝑦⟩ · 𝑎) 1 ) = 𝑟))
123	65, 75, 122	mp2and 715	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ (𝑦𝐻𝑎))) → (𝑟(⟨𝑦, 𝑦⟩ · 𝑎) 1 ) = 𝑟)
124	67	3ad2ant1 1082	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → ∀𝑦 ∈ 𝐵 (𝑦𝐻𝑦) ≠ ∅)
125		simp23 1096	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → 𝑧 ∈ 𝐵)
126		id 22	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
127	126, 126	oveq12d 6668	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑦𝐻𝑦) = (𝑧𝐻𝑧))
128	127	neeq1d 2853	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑦𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑧) ≠ ∅))
129	128	rspccva 3308	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝐵 (𝑦𝐻𝑦) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → (𝑧𝐻𝑧) ≠ ∅)
130	124, 125, 129	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → (𝑧𝐻𝑧) ≠ ∅)
131		n0 3931	. . . . 5 ⊢ ((𝑧𝐻𝑧) ≠ ∅ ↔ ∃𝑘 𝑘 ∈ (𝑧𝐻𝑧))
132	130, 131	sylib 208	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → ∃𝑘 𝑘 ∈ (𝑧𝐻𝑧))
133		eleq1 2689	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
134	133	3anbi1d 1403	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)))
135		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝑥𝐻𝑎) = (𝑦𝐻𝑎))
136	135	eleq2d 2687	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑟 ∈ (𝑥𝐻𝑎) ↔ 𝑟 ∈ (𝑦𝐻𝑎)))
137	136	anbi1d 741	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ↔ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))))
138	137	anbi1d 741	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)) ↔ ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧))))
139	134, 138	anbi12d 747	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧))) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))))
140	139	anbi2d 740	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) ↔ (𝜑 ∧ ((𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧))))))
141		opeq1 4402	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑎⟩ = ⟨𝑦, 𝑎⟩)
142	141	oveq1d 6665	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (⟨𝑥, 𝑎⟩ · 𝑧) = (⟨𝑦, 𝑎⟩ · 𝑧))
143	142	oveqd 6667	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟) = (𝑔(⟨𝑦, 𝑎⟩ · 𝑧)𝑟))
144		oveq1 6657	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥𝐻𝑧) = (𝑦𝐻𝑧))
145	143, 144	eleq12d 2695	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟) ∈ (𝑥𝐻𝑧) ↔ (𝑔(⟨𝑦, 𝑎⟩ · 𝑧)𝑟) ∈ (𝑦𝐻𝑧)))
146	140, 145	imbi12d 334	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) → (𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟) ∈ (𝑥𝐻𝑧)) ↔ ((𝜑 ∧ ((𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) → (𝑔(⟨𝑦, 𝑎⟩ · 𝑧)𝑟) ∈ (𝑦𝐻𝑧))))
147		df-3an 1039	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
148	22, 147	bitri 264	. . . . . . . . . . . . . 14 ⊢ (𝜓 ↔ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
149		simpll 790	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → 𝑦 = 𝑎)
150	149	eleq1d 2686	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑦 ∈ 𝐵 ↔ 𝑎 ∈ 𝐵))
151	150	anbi2d 740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵)))
152		simplr 792	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → 𝑤 = 𝑧)
153	152	eleq1d 2686	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑤 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
154	153	anbi2d 740	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ↔ (𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)))
155		anidm 676	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ↔ 𝑧 ∈ 𝐵)
156	154, 155	syl6bb 276	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ↔ 𝑧 ∈ 𝐵))
157	151, 156	anbi12d 747	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵)))
158		df-3an 1039	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵))
159	157, 158	syl6bbr 278	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)))
160		simpr 477	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → 𝑓 = 𝑟)
161	149	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑥𝐻𝑦) = (𝑥𝐻𝑎))
162	160, 161	eleq12d 2695	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑟 ∈ (𝑥𝐻𝑎)))
163	149	oveq1d 6665	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑦𝐻𝑧) = (𝑎𝐻𝑧))
164	163	eleq2d 2687	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑔 ∈ (𝑦𝐻𝑧) ↔ 𝑔 ∈ (𝑎𝐻𝑧)))
165	152	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑧𝐻𝑤) = (𝑧𝐻𝑧))
166	165	eleq2d 2687	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑘 ∈ (𝑧𝐻𝑤) ↔ 𝑘 ∈ (𝑧𝐻𝑧)))
167	162, 164, 166	3anbi123d 1399	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑧))))
168		df-3an 1039	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑧)) ↔ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))
169	167, 168	syl6bb 276	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧))))
170	159, 169	anbi12d 747	. . . . . . . . . . . . . 14 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))))
171	148, 170	syl5bb 272	. . . . . . . . . . . . 13 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝜓 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))))
172	171	anbi2d 740	. . . . . . . . . . . 12 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧))))))
173	149	opeq2d 4409	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑎⟩)
174	173	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (⟨𝑥, 𝑦⟩ · 𝑧) = (⟨𝑥, 𝑎⟩ · 𝑧))
175		eqidd 2623	. . . . . . . . . . . . . 14 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → 𝑔 = 𝑔)
176	174, 175, 160	oveq123d 6671	. . . . . . . . . . . . 13 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟))
177	176	eleq1d 2686	. . . . . . . . . . . 12 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ↔ (𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟) ∈ (𝑥𝐻𝑧)))
178	172, 177	imbi12d 334	. . . . . . . . . . 11 ⊢ (((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (((𝜑 ∧ 𝜓) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) ↔ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) → (𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟) ∈ (𝑥𝐻𝑧))))
179	178	sbiedv 2410	. . . . . . . . . 10 ⊢ ((𝑦 = 𝑎 ∧ 𝑤 = 𝑧) → ([𝑟 / 𝑓]((𝜑 ∧ 𝜓) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) ↔ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) → (𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟) ∈ (𝑥𝐻𝑧))))
180	179	sbiedv 2410	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → ([𝑧 / 𝑤][𝑟 / 𝑓]((𝜑 ∧ 𝜓) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) ↔ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) → (𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟) ∈ (𝑥𝐻𝑧))))
181		iscatd2.4	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
182	181	sbt 2419	. . . . . . . . . 10 ⊢ [𝑟 / 𝑓]((𝜑 ∧ 𝜓) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
183	182	sbt 2419	. . . . . . . . 9 ⊢ [𝑧 / 𝑤][𝑟 / 𝑓]((𝜑 ∧ 𝜓) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
184	180, 183	chvarv 2263	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) → (𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟) ∈ (𝑥𝐻𝑧))
185	146, 184	chvarv 2263	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) → (𝑔(⟨𝑦, 𝑎⟩ · 𝑧)𝑟) ∈ (𝑦𝐻𝑧))
186	185	exp45 642	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) → (𝑘 ∈ (𝑧𝐻𝑧) → (𝑔(⟨𝑦, 𝑎⟩ · 𝑧)𝑟) ∈ (𝑦𝐻𝑧)))))
187	186	3imp 1256	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → (𝑘 ∈ (𝑧𝐻𝑧) → (𝑔(⟨𝑦, 𝑎⟩ · 𝑧)𝑟) ∈ (𝑦𝐻𝑧)))
188	187	exlimdv 1861	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → (∃𝑘 𝑘 ∈ (𝑧𝐻𝑧) → (𝑔(⟨𝑦, 𝑎⟩ · 𝑧)𝑟) ∈ (𝑦𝐻𝑧)))
189	132, 188	mpd 15	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → (𝑔(⟨𝑦, 𝑎⟩ · 𝑧)𝑟) ∈ (𝑦𝐻𝑧))
190	133	anbi1d 741	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵)))
191	190	anbi1d 741	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))))
192	136	3anbi1d 1403	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
193	191, 192	3anbi23d 1402	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (𝜑 ∧ ((𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
194	141	oveq1d 6665	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (⟨𝑥, 𝑎⟩ · 𝑤) = (⟨𝑦, 𝑎⟩ · 𝑤))
195	194	oveqd 6667	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑘(⟨𝑎, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑎⟩ · 𝑤)𝑟) = ((𝑘(⟨𝑎, 𝑧⟩ · 𝑤)𝑔)(⟨𝑦, 𝑎⟩ · 𝑤)𝑟))
196		opeq1 4402	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
197	196	oveq1d 6665	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (⟨𝑥, 𝑧⟩ · 𝑤) = (⟨𝑦, 𝑧⟩ · 𝑤))
198		eqidd 2623	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝑘 = 𝑘)
199	197, 198, 143	oveq123d 6671	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟)) = (𝑘(⟨𝑦, 𝑧⟩ · 𝑤)(𝑔(⟨𝑦, 𝑎⟩ · 𝑧)𝑟)))
200	195, 199	eqeq12d 2637	. . . . 5 ⊢ (𝑥 = 𝑦 → (((𝑘(⟨𝑎, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑎⟩ · 𝑤)𝑟) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟)) ↔ ((𝑘(⟨𝑎, 𝑧⟩ · 𝑤)𝑔)(⟨𝑦, 𝑎⟩ · 𝑤)𝑟) = (𝑘(⟨𝑦, 𝑧⟩ · 𝑤)(𝑔(⟨𝑦, 𝑎⟩ · 𝑧)𝑟))))
201	193, 200	imbi12d 334	. . . 4 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑎, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑎⟩ · 𝑤)𝑟) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟))) ↔ ((𝜑 ∧ ((𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑎, 𝑧⟩ · 𝑤)𝑔)(⟨𝑦, 𝑎⟩ · 𝑤)𝑟) = (𝑘(⟨𝑦, 𝑧⟩ · 𝑤)(𝑔(⟨𝑦, 𝑎⟩ · 𝑧)𝑟)))))
202		simpl 473	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → 𝑦 = 𝑎)
203	202	eleq1d 2686	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → (𝑦 ∈ 𝐵 ↔ 𝑎 ∈ 𝐵))
204	203	anbi2d 740	. . . . . . . . . . . 12 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵)))
205		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → 𝑓 = 𝑟)
206	202	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → (𝑥𝐻𝑦) = (𝑥𝐻𝑎))
207	205, 206	eleq12d 2695	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑟 ∈ (𝑥𝐻𝑎)))
208	202	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → (𝑦𝐻𝑧) = (𝑎𝐻𝑧))
209	208	eleq2d 2687	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → (𝑔 ∈ (𝑦𝐻𝑧) ↔ 𝑔 ∈ (𝑎𝐻𝑧)))
210	207, 209	3anbi12d 1400	. . . . . . . . . . . 12 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
211	204, 210	3anbi13d 1401	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
212	22, 211	syl5bb 272	. . . . . . . . . 10 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → (𝜓 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
213		df-3an 1039	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
214	212, 213	syl6bb 276	. . . . . . . . 9 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → (𝜓 ↔ (((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
215	214	anbi2d 740	. . . . . . . 8 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))))
216		3anass 1042	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (𝜑 ∧ (((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
217	215, 216	syl6bbr 278	. . . . . . 7 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
218	202	opeq2d 4409	. . . . . . . . . 10 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑎⟩)
219	218	oveq1d 6665	. . . . . . . . 9 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → (⟨𝑥, 𝑦⟩ · 𝑤) = (⟨𝑥, 𝑎⟩ · 𝑤))
220	202	opeq1d 4408	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → ⟨𝑦, 𝑧⟩ = ⟨𝑎, 𝑧⟩)
221	220	oveq1d 6665	. . . . . . . . . 10 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → (⟨𝑦, 𝑧⟩ · 𝑤) = (⟨𝑎, 𝑧⟩ · 𝑤))
222	221	oveqd 6667	. . . . . . . . 9 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → (𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔) = (𝑘(⟨𝑎, 𝑧⟩ · 𝑤)𝑔))
223	219, 222, 205	oveq123d 6671	. . . . . . . 8 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = ((𝑘(⟨𝑎, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑎⟩ · 𝑤)𝑟))
224	218	oveq1d 6665	. . . . . . . . . 10 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → (⟨𝑥, 𝑦⟩ · 𝑧) = (⟨𝑥, 𝑎⟩ · 𝑧))
225		eqidd 2623	. . . . . . . . . 10 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → 𝑔 = 𝑔)
226	224, 225, 205	oveq123d 6671	. . . . . . . . 9 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟))
227	226	oveq2d 6666	. . . . . . . 8 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟)))
228	223, 227	eqeq12d 2637	. . . . . . 7 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → (((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)) ↔ ((𝑘(⟨𝑎, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑎⟩ · 𝑤)𝑟) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟))))
229	217, 228	imbi12d 334	. . . . . 6 ⊢ ((𝑦 = 𝑎 ∧ 𝑓 = 𝑟) → (((𝜑 ∧ 𝜓) → ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))) ↔ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑎, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑎⟩ · 𝑤)𝑟) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟)))))
230	229	sbiedv 2410	. . . . 5 ⊢ (𝑦 = 𝑎 → ([𝑟 / 𝑓]((𝜑 ∧ 𝜓) → ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))) ↔ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑎, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑎⟩ · 𝑤)𝑟) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟)))))
231		iscatd2.5	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))
232	231	sbt 2419	. . . . 5 ⊢ [𝑟 / 𝑓]((𝜑 ∧ 𝜓) → ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))
233	230, 232	chvarv 2263	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑎, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑎⟩ · 𝑤)𝑟) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑎⟩ · 𝑧)𝑟)))
234	201, 233	chvarv 2263	. . 3 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑎, 𝑧⟩ · 𝑤)𝑔)(⟨𝑦, 𝑎⟩ · 𝑤)𝑟) = (𝑘(⟨𝑦, 𝑧⟩ · 𝑤)(𝑔(⟨𝑦, 𝑎⟩ · 𝑧)𝑟)))
235	1, 2, 3, 4, 5, 62, 123, 189, 234	iscatd 16334	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
236	1, 2, 3, 235, 5, 62, 123	catidd 16341	. 2 ⊢ (𝜑 → (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 1 ))
237	235, 236	jca 554	1 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 1 )))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704  [wsb 1880   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∅c0 3915  ⟨cop 4183   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Hom chom 15952  compcco 15953  Catccat 16325  Idccid 16326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-cat 16329  df-cid 16330

This theorem is referenced by:  oppccatid  16379  subccatid  16506  fuccatid  16629  setccatid  16734  catccatid  16752  estrccatid  16772  xpccatid  16828  rngccatidALTV  41989  ringccatidALTV  42052