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Theorem mdetunilem7 22112

Description: Lemma for mdetuni 22116. (Contributed by SO, 15-Jul-2018.)

**Hypotheses**
Ref	Expression
mdetuni.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mdetuni.b	⊢ 𝐵 = (Base‘𝐴)
mdetuni.k	⊢ 𝐾 = (Base‘𝑅)
mdetuni.0g	⊢ 0 = (0_g‘𝑅)
mdetuni.1r	⊢ 1 = (1_r‘𝑅)
mdetuni.pg	⊢ + = (+_g‘𝑅)
mdetuni.tg	⊢ · = (._r‘𝑅)
mdetuni.n	⊢ (𝜑 → 𝑁 ∈ Fin)
mdetuni.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mdetuni.ff	⊢ (𝜑 → 𝐷:𝐵⟶𝐾)
mdetuni.al	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))
mdetuni.li	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘_f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))
mdetuni.sc	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘_f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))

**Assertion**
Ref	Expression
mdetunilem7	⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘𝐹)))

Distinct variable groups: 𝜑,𝑥,𝑦,𝑧,𝑤,𝑎,𝑏 𝑥,𝐵,𝑦,𝑧,𝑤,𝑎,𝑏 𝑥,𝐾,𝑦,𝑧,𝑤,𝑎,𝑏 𝑥,𝑁,𝑦,𝑧,𝑤,𝑎,𝑏 𝑥,𝐷,𝑦,𝑧,𝑤,𝑎,𝑏 𝑥, · ,𝑦,𝑧,𝑤 + ,𝑎,𝑏,𝑥,𝑦,𝑧,𝑤 0 ,𝑎,𝑏,𝑥,𝑦,𝑧,𝑤 1 ,𝑎,𝑏,𝑥,𝑦,𝑧,𝑤 𝑥,𝑅,𝑦,𝑧,𝑤 𝐴,𝑎,𝑏,𝑥,𝑦,𝑧,𝑤 𝑥,𝐸,𝑦,𝑧,𝑤 𝑥,𝐹,𝑦,𝑧,𝑤 𝐸,𝑎,𝑏 𝐹,𝑎,𝑏 Allowed substitution hints: 𝑅(𝑎,𝑏) · (𝑎,𝑏)

Proof of Theorem mdetunilem7 Dummy variables 𝑐 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6888	. . . . . 6 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑎) = (𝑑‘𝑎))
2	1	oveq1d 7421	. . . . 5 ⊢ (𝑐 = 𝑑 → ((𝑐‘𝑎)𝐹𝑏) = ((𝑑‘𝑎)𝐹𝑏))
3	2	mpoeq3dv 7485	. . . 4 ⊢ (𝑐 = 𝑑 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏)))
4	3	fveq2d 6893	. . 3 ⊢ (𝑐 = 𝑑 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏))))
5		fveq2 6889	. . . 4 ⊢ (𝑐 = 𝑑 → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) = (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑))
6	5	oveq1d 7421	. . 3 ⊢ (𝑐 = 𝑑 → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) · (𝐷‘𝐹)) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (𝐷‘𝐹)))
7	4, 6	eqeq12d 2749	. 2 ⊢ (𝑐 = 𝑑 → ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) · (𝐷‘𝐹)) ↔ (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (𝐷‘𝐹))))
8		fveq1 6888	. . . . . 6 ⊢ (𝑐 = (𝑑(+_g‘(SymGrp‘𝑁))𝑒) → (𝑐‘𝑎) = ((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎))
9	8	oveq1d 7421	. . . . 5 ⊢ (𝑐 = (𝑑(+_g‘(SymGrp‘𝑁))𝑒) → ((𝑐‘𝑎)𝐹𝑏) = (((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎)𝐹𝑏))
10	9	mpoeq3dv 7485	. . . 4 ⊢ (𝑐 = (𝑑(+_g‘(SymGrp‘𝑁))𝑒) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎)𝐹𝑏)))
11	10	fveq2d 6893	. . 3 ⊢ (𝑐 = (𝑑(+_g‘(SymGrp‘𝑁))𝑒) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎)𝐹𝑏))))
12		fveq2 6889	. . . 4 ⊢ (𝑐 = (𝑑(+_g‘(SymGrp‘𝑁))𝑒) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) = (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(𝑑(+_g‘(SymGrp‘𝑁))𝑒)))
13	12	oveq1d 7421	. . 3 ⊢ (𝑐 = (𝑑(+_g‘(SymGrp‘𝑁))𝑒) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) · (𝐷‘𝐹)) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(𝑑(+_g‘(SymGrp‘𝑁))𝑒)) · (𝐷‘𝐹)))
14	11, 13	eqeq12d 2749	. 2 ⊢ (𝑐 = (𝑑(+_g‘(SymGrp‘𝑁))𝑒) → ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) · (𝐷‘𝐹)) ↔ (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(𝑑(+_g‘(SymGrp‘𝑁))𝑒)) · (𝐷‘𝐹))))
15		fveq1 6888	. . . . . 6 ⊢ (𝑐 = (0_g‘(SymGrp‘𝑁)) → (𝑐‘𝑎) = ((0_g‘(SymGrp‘𝑁))‘𝑎))
16	15	oveq1d 7421	. . . . 5 ⊢ (𝑐 = (0_g‘(SymGrp‘𝑁)) → ((𝑐‘𝑎)𝐹𝑏) = (((0_g‘(SymGrp‘𝑁))‘𝑎)𝐹𝑏))
17	16	mpoeq3dv 7485	. . . 4 ⊢ (𝑐 = (0_g‘(SymGrp‘𝑁)) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((0_g‘(SymGrp‘𝑁))‘𝑎)𝐹𝑏)))
18	17	fveq2d 6893	. . 3 ⊢ (𝑐 = (0_g‘(SymGrp‘𝑁)) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((0_g‘(SymGrp‘𝑁))‘𝑎)𝐹𝑏))))
19		fveq2 6889	. . . 4 ⊢ (𝑐 = (0_g‘(SymGrp‘𝑁)) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) = (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(0_g‘(SymGrp‘𝑁))))
20	19	oveq1d 7421	. . 3 ⊢ (𝑐 = (0_g‘(SymGrp‘𝑁)) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) · (𝐷‘𝐹)) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(0_g‘(SymGrp‘𝑁))) · (𝐷‘𝐹)))
21	18, 20	eqeq12d 2749	. 2 ⊢ (𝑐 = (0_g‘(SymGrp‘𝑁)) → ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) · (𝐷‘𝐹)) ↔ (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((0_g‘(SymGrp‘𝑁))‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(0_g‘(SymGrp‘𝑁))) · (𝐷‘𝐹))))
22		fveq1 6888	. . . . . 6 ⊢ (𝑐 = 𝐸 → (𝑐‘𝑎) = (𝐸‘𝑎))
23	22	oveq1d 7421	. . . . 5 ⊢ (𝑐 = 𝐸 → ((𝑐‘𝑎)𝐹𝑏) = ((𝐸‘𝑎)𝐹𝑏))
24	23	mpoeq3dv 7485	. . . 4 ⊢ (𝑐 = 𝐸 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)𝐹𝑏)))
25	24	fveq2d 6893	. . 3 ⊢ (𝑐 = 𝐸 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)𝐹𝑏))))
26		fveq2 6889	. . . 4 ⊢ (𝑐 = 𝐸 → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) = (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸))
27	26	oveq1d 7421	. . 3 ⊢ (𝑐 = 𝐸 → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) · (𝐷‘𝐹)) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘𝐹)))
28	25, 27	eqeq12d 2749	. 2 ⊢ (𝑐 = 𝐸 → ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) · (𝐷‘𝐹)) ↔ (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘𝐹))))
29		eqid 2733	. 2 ⊢ (0_g‘(SymGrp‘𝑁)) = (0_g‘(SymGrp‘𝑁))
30		eqid 2733	. 2 ⊢ (+_g‘(SymGrp‘𝑁)) = (+_g‘(SymGrp‘𝑁))
31		eqid 2733	. 2 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
32		mdetuni.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ Fin)
33	32	3ad2ant1 1134	. . 3 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝑁 ∈ Fin)
34		eqid 2733	. . . 4 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
35	34	symggrp 19263	. . 3 ⊢ (𝑁 ∈ Fin → (SymGrp‘𝑁) ∈ Grp)
36		grpmnd 18823	. . 3 ⊢ ((SymGrp‘𝑁) ∈ Grp → (SymGrp‘𝑁) ∈ Mnd)
37	33, 35, 36	3syl 18	. 2 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (SymGrp‘𝑁) ∈ Mnd)
38		eqid 2733	. . . 4 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁)
39	38, 34, 31	symgtrf 19332	. . 3 ⊢ ran (pmTrsp‘𝑁) ⊆ (Base‘(SymGrp‘𝑁))
40	39	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → ran (pmTrsp‘𝑁) ⊆ (Base‘(SymGrp‘𝑁)))
41		eqid 2733	. . . . . 6 ⊢ (mrCls‘(SubMnd‘(SymGrp‘𝑁))) = (mrCls‘(SubMnd‘(SymGrp‘𝑁)))
42	38, 34, 31, 41	symggen2 19334	. . . . 5 ⊢ (𝑁 ∈ Fin → ((mrCls‘(SubMnd‘(SymGrp‘𝑁)))‘ran (pmTrsp‘𝑁)) = (Base‘(SymGrp‘𝑁)))
43	32, 42	syl 17	. . . 4 ⊢ (𝜑 → ((mrCls‘(SubMnd‘(SymGrp‘𝑁)))‘ran (pmTrsp‘𝑁)) = (Base‘(SymGrp‘𝑁)))
44	43	eqcomd 2739	. . 3 ⊢ (𝜑 → (Base‘(SymGrp‘𝑁)) = ((mrCls‘(SubMnd‘(SymGrp‘𝑁)))‘ran (pmTrsp‘𝑁)))
45	44	3ad2ant1 1134	. 2 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (Base‘(SymGrp‘𝑁)) = ((mrCls‘(SubMnd‘(SymGrp‘𝑁)))‘ran (pmTrsp‘𝑁)))
46		mdetuni.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
47	46	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝑅 ∈ Ring)
48		mdetuni.ff	. . . . . 6 ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)
49	48	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐷:𝐵⟶𝐾)
50		simp3 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐹 ∈ 𝐵)
51	49, 50	ffvelcdmd 7085	. . . 4 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (𝐷‘𝐹) ∈ 𝐾)
52		mdetuni.k	. . . . 5 ⊢ 𝐾 = (Base‘𝑅)
53		mdetuni.tg	. . . . 5 ⊢ · = (._r‘𝑅)
54		mdetuni.1r	. . . . 5 ⊢ 1 = (1_r‘𝑅)
55	52, 53, 54	ringlidm 20080	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐷‘𝐹) ∈ 𝐾) → ( 1 · (𝐷‘𝐹)) = (𝐷‘𝐹))
56	47, 51, 55	syl2anc 585	. . 3 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → ( 1 · (𝐷‘𝐹)) = (𝐷‘𝐹))
57		zrhpsgnmhm 21129	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
58	46, 32, 57	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
59		eqid 2733	. . . . . . . 8 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
60	59, 54	ringidval 20001	. . . . . . 7 ⊢ 1 = (0_g‘(mulGrp‘𝑅))
61	29, 60	mhm0 18677	. . . . . 6 ⊢ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(0_g‘(SymGrp‘𝑁))) = 1 )
62	58, 61	syl 17	. . . . 5 ⊢ (𝜑 → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(0_g‘(SymGrp‘𝑁))) = 1 )
63	62	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(0_g‘(SymGrp‘𝑁))) = 1 )
64	63	oveq1d 7421	. . 3 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(0_g‘(SymGrp‘𝑁))) · (𝐷‘𝐹)) = ( 1 · (𝐷‘𝐹)))
65	34	symgid 19264	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ Fin → ( I ↾ 𝑁) = (0_g‘(SymGrp‘𝑁)))
66	32, 65	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ( I ↾ 𝑁) = (0_g‘(SymGrp‘𝑁)))
67	66	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → ( I ↾ 𝑁) = (0_g‘(SymGrp‘𝑁)))
68	67	3ad2ant1 1134	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ( I ↾ 𝑁) = (0_g‘(SymGrp‘𝑁)))
69	68	fveq1d 6891	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (( I ↾ 𝑁)‘𝑎) = ((0_g‘(SymGrp‘𝑁))‘𝑎))
70		simp2 1138	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑎 ∈ 𝑁)
71		fvresi 7168	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝑁 → (( I ↾ 𝑁)‘𝑎) = 𝑎)
72	70, 71	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (( I ↾ 𝑁)‘𝑎) = 𝑎)
73	69, 72	eqtr3d 2775	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((0_g‘(SymGrp‘𝑁))‘𝑎) = 𝑎)
74	73	oveq1d 7421	. . . . . 6 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (((0_g‘(SymGrp‘𝑁))‘𝑎)𝐹𝑏) = (𝑎𝐹𝑏))
75	74	mpoeq3dva 7483	. . . . 5 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((0_g‘(SymGrp‘𝑁))‘𝑎)𝐹𝑏)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (𝑎𝐹𝑏)))
76		mdetuni.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅)
77		mdetuni.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐴)
78	76, 52, 77	matbas2i 21916	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
79	78	3ad2ant3 1136	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐹 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
80		elmapi 8840	. . . . . . 7 ⊢ (𝐹 ∈ (𝐾 ↑_m (𝑁 × 𝑁)) → 𝐹:(𝑁 × 𝑁)⟶𝐾)
81		ffn 6715	. . . . . . 7 ⊢ (𝐹:(𝑁 × 𝑁)⟶𝐾 → 𝐹 Fn (𝑁 × 𝑁))
82	79, 80, 81	3syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐹 Fn (𝑁 × 𝑁))
83		fnov 7537	. . . . . 6 ⊢ (𝐹 Fn (𝑁 × 𝑁) ↔ 𝐹 = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (𝑎𝐹𝑏)))
84	82, 83	sylib 217	. . . . 5 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐹 = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (𝑎𝐹𝑏)))
85	75, 84	eqtr4d 2776	. . . 4 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((0_g‘(SymGrp‘𝑁))‘𝑎)𝐹𝑏)) = 𝐹)
86	85	fveq2d 6893	. . 3 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((0_g‘(SymGrp‘𝑁))‘𝑎)𝐹𝑏))) = (𝐷‘𝐹))
87	56, 64, 86	3eqtr4rd 2784	. 2 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((0_g‘(SymGrp‘𝑁))‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(0_g‘(SymGrp‘𝑁))) · (𝐷‘𝐹)))
88		simp2 1138	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑑 ∈ (Base‘(SymGrp‘𝑁)))
89	39	sseli 3978	. . . . . . . . . . . . 13 ⊢ (𝑒 ∈ ran (pmTrsp‘𝑁) → 𝑒 ∈ (Base‘(SymGrp‘𝑁)))
90	89	3ad2ant3 1136	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑒 ∈ (Base‘(SymGrp‘𝑁)))
91	34, 31, 30	symgov 19246	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ (Base‘(SymGrp‘𝑁))) → (𝑑(+_g‘(SymGrp‘𝑁))𝑒) = (𝑑 ∘ 𝑒))
92	88, 90, 91	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → (𝑑(+_g‘(SymGrp‘𝑁))𝑒) = (𝑑 ∘ 𝑒))
93	92	fveq1d 6891	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → ((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎) = ((𝑑 ∘ 𝑒)‘𝑎))
94	93	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎) = ((𝑑 ∘ 𝑒)‘𝑎))
95	34, 31	symgbasf1o 19237	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ (Base‘(SymGrp‘𝑁)) → 𝑒:𝑁–1-1-onto→𝑁)
96		f1of 6831	. . . . . . . . . . . 12 ⊢ (𝑒:𝑁–1-1-onto→𝑁 → 𝑒:𝑁⟶𝑁)
97	90, 95, 96	3syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑒:𝑁⟶𝑁)
98	97	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑒:𝑁⟶𝑁)
99		simp2 1138	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑎 ∈ 𝑁)
100		fvco3 6988	. . . . . . . . . 10 ⊢ ((𝑒:𝑁⟶𝑁 ∧ 𝑎 ∈ 𝑁) → ((𝑑 ∘ 𝑒)‘𝑎) = (𝑑‘(𝑒‘𝑎)))
101	98, 99, 100	syl2anc 585	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑑 ∘ 𝑒)‘𝑎) = (𝑑‘(𝑒‘𝑎)))
102	94, 101	eqtrd 2773	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎) = (𝑑‘(𝑒‘𝑎)))
103	102	oveq1d 7421	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎)𝐹𝑏) = ((𝑑‘(𝑒‘𝑎))𝐹𝑏))
104	103	mpoeq3dva 7483	. . . . . 6 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎)𝐹𝑏)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(𝑒‘𝑎))𝐹𝑏)))
105	104	fveq2d 6893	. . . . 5 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎)𝐹𝑏))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(𝑒‘𝑎))𝐹𝑏))))
106	34, 31	symgbasf 19238	. . . . . 6 ⊢ (𝑑 ∈ (Base‘(SymGrp‘𝑁)) → 𝑑:𝑁⟶𝑁)
107		eqid 2733	. . . . . . . . 9 ⊢ (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
108	107, 38	pmtrrn2 19323	. . . . . . . 8 ⊢ (𝑒 ∈ ran (pmTrsp‘𝑁) → ∃𝑐 ∈ 𝑁 ∃𝑓 ∈ 𝑁 (𝑐 ≠ 𝑓 ∧ 𝑒 = ((pmTrsp‘𝑁)‘{𝑐, 𝑓})))
109		mdetuni.0g	. . . . . . . . . . . . . 14 ⊢ 0 = (0_g‘𝑅)
110		mdetuni.pg	. . . . . . . . . . . . . 14 ⊢ + = (+_g‘𝑅)
111		mdetuni.al	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))
112		mdetuni.li	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘_f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))
113		mdetuni.sc	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘_f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))
114		simpll1 1213	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → 𝜑)
115		df-3an 1090	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁 ∧ 𝑐 ≠ 𝑓) ↔ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓))
116	115	biimpri 227	. . . . . . . . . . . . . . 15 ⊢ (((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓) → (𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁 ∧ 𝑐 ≠ 𝑓))
117	116	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → (𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁 ∧ 𝑐 ≠ 𝑓))
118	79, 80	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐹:(𝑁 × 𝑁)⟶𝐾)
119	118	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) → 𝐹:(𝑁 × 𝑁)⟶𝐾)
120	119	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑏 ∈ 𝑁) → 𝐹:(𝑁 × 𝑁)⟶𝐾)
121		simpllr 775	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑏 ∈ 𝑁) → 𝑑:𝑁⟶𝑁)
122		simprlr 779	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → 𝑓 ∈ 𝑁)
123	122	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑏 ∈ 𝑁) → 𝑓 ∈ 𝑁)
124	121, 123	ffvelcdmd 7085	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑏 ∈ 𝑁) → (𝑑‘𝑓) ∈ 𝑁)
125		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑏 ∈ 𝑁) → 𝑏 ∈ 𝑁)
126	120, 124, 125	fovcdmd 7576	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑏 ∈ 𝑁) → ((𝑑‘𝑓)𝐹𝑏) ∈ 𝐾)
127		simprll 778	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → 𝑐 ∈ 𝑁)
128	127	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑏 ∈ 𝑁) → 𝑐 ∈ 𝑁)
129	121, 128	ffvelcdmd 7085	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑏 ∈ 𝑁) → (𝑑‘𝑐) ∈ 𝑁)
130	120, 129, 125	fovcdmd 7576	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑏 ∈ 𝑁) → ((𝑑‘𝑐)𝐹𝑏) ∈ 𝐾)
131	126, 130	jca 513	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑏 ∈ 𝑁) → (((𝑑‘𝑓)𝐹𝑏) ∈ 𝐾 ∧ ((𝑑‘𝑐)𝐹𝑏) ∈ 𝐾))
132	118	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → 𝐹:(𝑁 × 𝑁)⟶𝐾)
133	132	3ad2ant1 1134	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐹:(𝑁 × 𝑁)⟶𝐾)
134		simp1lr 1238	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑑:𝑁⟶𝑁)
135		simp2 1138	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑎 ∈ 𝑁)
136	134, 135	ffvelcdmd 7085	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑑‘𝑎) ∈ 𝑁)
137		simp3 1139	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑏 ∈ 𝑁)
138	133, 136, 137	fovcdmd 7576	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑑‘𝑎)𝐹𝑏) ∈ 𝐾)
139	76, 77, 52, 109, 54, 110, 53, 32, 46, 48, 111, 112, 113, 114, 117, 131, 138	mdetunilem6 22111	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, ((𝑑‘𝑓)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, ((𝑑‘𝑐)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))))))
140		simpl1 1192	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) → 𝜑)
141		fveq2 6889	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑐 → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑐))
142	32	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → 𝑁 ∈ Fin)
143		simprll 778	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → 𝑐 ∈ 𝑁)
144		simprlr 779	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → 𝑓 ∈ 𝑁)
145		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → 𝑐 ≠ 𝑓)
146	107	pmtrprfv 19316	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ Fin ∧ (𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁 ∧ 𝑐 ≠ 𝑓)) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑐) = 𝑓)
147	142, 143, 144, 145, 146	syl13anc 1373	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑐) = 𝑓)
148	147	adantr 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑐) = 𝑓)
149	141, 148	sylan9eqr 2795	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑐) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = 𝑓)
150	149	fveq2d 6893	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑐) → (𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎)) = (𝑑‘𝑓))
151	150	oveq1d 7421	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑐) → ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏) = ((𝑑‘𝑓)𝐹𝑏))
152		iftrue 4534	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑐 → if(𝑎 = 𝑐, ((𝑑‘𝑓)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))) = ((𝑑‘𝑓)𝐹𝑏))
153	152	adantl 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑐) → if(𝑎 = 𝑐, ((𝑑‘𝑓)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))) = ((𝑑‘𝑓)𝐹𝑏))
154	151, 153	eqtr4d 2776	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑐) → ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏) = if(𝑎 = 𝑐, ((𝑑‘𝑓)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))))
155		fveq2 6889	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑓 → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑓))
156		prcom 4736	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ {𝑐, 𝑓} = {𝑓, 𝑐}
157	156	fveq2i 6892	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) = ((pmTrsp‘𝑁)‘{𝑓, 𝑐})
158	157	fveq1i 6890	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑓) = (((pmTrsp‘𝑁)‘{𝑓, 𝑐})‘𝑓)
159	32	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ Fin)
160		simplrl 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → (𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁))
161	160	simprd 497	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → 𝑓 ∈ 𝑁)
162	160	simpld 496	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → 𝑐 ∈ 𝑁)
163		simplrr 777	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → 𝑐 ≠ 𝑓)
164	163	necomd 2997	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → 𝑓 ≠ 𝑐)
165	107	pmtrprfv 19316	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑁 ∈ Fin ∧ (𝑓 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁 ∧ 𝑓 ≠ 𝑐)) → (((pmTrsp‘𝑁)‘{𝑓, 𝑐})‘𝑓) = 𝑐)
166	159, 161, 162, 164, 165	syl13anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → (((pmTrsp‘𝑁)‘{𝑓, 𝑐})‘𝑓) = 𝑐)
167	158, 166	eqtrid 2785	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑓) = 𝑐)
168	155, 167	sylan9eqr 2795	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑓) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = 𝑐)
169	168	fveq2d 6893	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑓) → (𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎)) = (𝑑‘𝑐))
170	169	oveq1d 7421	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑓) → ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏) = ((𝑑‘𝑐)𝐹𝑏))
171		iftrue 4534	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑓 → if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)) = ((𝑑‘𝑐)𝐹𝑏))
172	171	adantl 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑓) → if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)) = ((𝑑‘𝑐)𝐹𝑏))
173	170, 172	eqtr4d 2776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑓) → ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
174	173	adantlr 714	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ 𝑎 = 𝑓) → ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
175		vex 3479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝑎 ∈ V
176	175	elpr 4651	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 ∈ {𝑐, 𝑓} ↔ (𝑎 = 𝑐 ∨ 𝑎 = 𝑓))
177	176	notbii 320	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (¬ 𝑎 ∈ {𝑐, 𝑓} ↔ ¬ (𝑎 = 𝑐 ∨ 𝑎 = 𝑓))
178		ioran 983	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (¬ (𝑎 = 𝑐 ∨ 𝑎 = 𝑓) ↔ (¬ 𝑎 = 𝑐 ∧ ¬ 𝑎 = 𝑓))
179	177, 178	sylbbr 235	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((¬ 𝑎 = 𝑐 ∧ ¬ 𝑎 = 𝑓) → ¬ 𝑎 ∈ {𝑐, 𝑓})
180	179	adantll 713	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ ¬ 𝑎 = 𝑓) → ¬ 𝑎 ∈ {𝑐, 𝑓})
181		prssi 4824	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) → {𝑐, 𝑓} ⊆ 𝑁)
182	160, 181	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → {𝑐, 𝑓} ⊆ 𝑁)
183		pr2ne 9996	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) → ({𝑐, 𝑓} ≈ 2_o ↔ 𝑐 ≠ 𝑓))
184	160, 183	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → ({𝑐, 𝑓} ≈ 2_o ↔ 𝑐 ≠ 𝑓))
185	163, 184	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → {𝑐, 𝑓} ≈ 2_o)
186	107	pmtrmvd 19319	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ Fin ∧ {𝑐, 𝑓} ⊆ 𝑁 ∧ {𝑐, 𝑓} ≈ 2_o) → dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I ) = {𝑐, 𝑓})
187	159, 182, 185, 186	syl3anc 1372	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I ) = {𝑐, 𝑓})
188	187	eleq2d 2820	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → (𝑎 ∈ dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I ) ↔ 𝑎 ∈ {𝑐, 𝑓}))
189	188	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → (¬ 𝑎 ∈ dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I ) ↔ ¬ 𝑎 ∈ {𝑐, 𝑓}))
190	189	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ ¬ 𝑎 = 𝑓) → (¬ 𝑎 ∈ dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I ) ↔ ¬ 𝑎 ∈ {𝑐, 𝑓}))
191	180, 190	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ ¬ 𝑎 = 𝑓) → ¬ 𝑎 ∈ dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I ))
192	107	pmtrf 19318	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 ∈ Fin ∧ {𝑐, 𝑓} ⊆ 𝑁 ∧ {𝑐, 𝑓} ≈ 2_o) → ((pmTrsp‘𝑁)‘{𝑐, 𝑓}):𝑁⟶𝑁)
193	159, 182, 185, 192	syl3anc 1372	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → ((pmTrsp‘𝑁)‘{𝑐, 𝑓}):𝑁⟶𝑁)
194	193	ffnd 6716	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) Fn 𝑁)
195		simpr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → 𝑎 ∈ 𝑁)
196		fnelnfp 7172	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((pmTrsp‘𝑁)‘{𝑐, 𝑓}) Fn 𝑁 ∧ 𝑎 ∈ 𝑁) → (𝑎 ∈ dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I ) ↔ (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) ≠ 𝑎))
197	196	necon2bbid 2985	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((pmTrsp‘𝑁)‘{𝑐, 𝑓}) Fn 𝑁 ∧ 𝑎 ∈ 𝑁) → ((((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = 𝑎 ↔ ¬ 𝑎 ∈ dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I )))
198	194, 195, 197	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → ((((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = 𝑎 ↔ ¬ 𝑎 ∈ dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I )))
199	198	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ ¬ 𝑎 = 𝑓) → ((((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = 𝑎 ↔ ¬ 𝑎 ∈ dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I )))
200	191, 199	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ ¬ 𝑎 = 𝑓) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = 𝑎)
201	200	fveq2d 6893	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ ¬ 𝑎 = 𝑓) → (𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎)) = (𝑑‘𝑎))
202	201	oveq1d 7421	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ ¬ 𝑎 = 𝑓) → ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏) = ((𝑑‘𝑎)𝐹𝑏))
203		iffalse 4537	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑎 = 𝑓 → if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)) = ((𝑑‘𝑎)𝐹𝑏))
204	203	adantl 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ ¬ 𝑎 = 𝑓) → if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)) = ((𝑑‘𝑎)𝐹𝑏))
205	202, 204	eqtr4d 2776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ ¬ 𝑎 = 𝑓) → ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
206	174, 205	pm2.61dan 812	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) → ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
207		iffalse 4537	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑎 = 𝑐 → if(𝑎 = 𝑐, ((𝑑‘𝑓)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))) = if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
208	207	adantl 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) → if(𝑎 = 𝑐, ((𝑑‘𝑓)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))) = if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
209	206, 208	eqtr4d 2776	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) → ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏) = if(𝑎 = 𝑐, ((𝑑‘𝑓)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))))
210	154, 209	pm2.61dan 812	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏) = if(𝑎 = 𝑐, ((𝑑‘𝑓)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))))
211	210	3adant3 1133	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏) = if(𝑎 = 𝑐, ((𝑑‘𝑓)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))))
212	211	mpoeq3dva 7483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, ((𝑑‘𝑓)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))))
213	140, 212	sylan 581	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, ((𝑑‘𝑓)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))))
214	213	fveq2d 6893	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, ((𝑑‘𝑓)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))))))
215		fveq2 6889	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑐 → (𝑑‘𝑎) = (𝑑‘𝑐))
216	215	oveq1d 7421	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑐 → ((𝑑‘𝑎)𝐹𝑏) = ((𝑑‘𝑐)𝐹𝑏))
217		iftrue 4534	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑐 → if(𝑎 = 𝑐, ((𝑑‘𝑐)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))) = ((𝑑‘𝑐)𝐹𝑏))
218	216, 217	eqtr4d 2776	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑐 → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑐, ((𝑑‘𝑐)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))))
219		fveq2 6889	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑓 → (𝑑‘𝑎) = (𝑑‘𝑓))
220	219	oveq1d 7421	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑓 → ((𝑑‘𝑎)𝐹𝑏) = ((𝑑‘𝑓)𝐹𝑏))
221		iftrue 4534	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑓 → if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)) = ((𝑑‘𝑓)𝐹𝑏))
222	220, 221	eqtr4d 2776	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑓 → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
223	222	adantl 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ 𝑎 = 𝑐 ∧ 𝑎 = 𝑓) → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
224		iffalse 4537	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝑎 = 𝑓 → if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)) = ((𝑑‘𝑎)𝐹𝑏))
225	224	eqcomd 2739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑎 = 𝑓 → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
226	225	adantl 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ 𝑎 = 𝑐 ∧ ¬ 𝑎 = 𝑓) → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
227	223, 226	pm2.61dan 812	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑎 = 𝑐 → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
228		iffalse 4537	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑎 = 𝑐 → if(𝑎 = 𝑐, ((𝑑‘𝑐)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))) = if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
229	227, 228	eqtr4d 2776	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑎 = 𝑐 → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑐, ((𝑑‘𝑐)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))))
230	218, 229	pm2.61i 182	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑐, ((𝑑‘𝑐)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
231	230	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑐, ((𝑑‘𝑐)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))))
232	231	mpoeq3ia 7484	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, ((𝑑‘𝑐)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))))
233	232	fveq2i 6892	. . . . . . . . . . . . . . 15 ⊢ (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, ((𝑑‘𝑐)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))))
234	233	fveq2i 6892	. . . . . . . . . . . . . 14 ⊢ ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏)))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, ((𝑑‘𝑐)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))))))
235	234	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏)))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, ((𝑑‘𝑐)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))))))
236	139, 214, 235	3eqtr4d 2783	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏)))))
237		fveq1 6888	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) → (𝑒‘𝑎) = (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))
238	237	fveq2d 6893	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) → (𝑑‘(𝑒‘𝑎)) = (𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎)))
239	238	oveq1d 7421	. . . . . . . . . . . . . 14 ⊢ (𝑒 = ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) → ((𝑑‘(𝑒‘𝑎))𝐹𝑏) = ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏))
240	239	mpoeq3dv 7485	. . . . . . . . . . . . 13 ⊢ (𝑒 = ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(𝑒‘𝑎))𝐹𝑏)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏)))
241	240	fveqeq2d 6897	. . . . . . . . . . . 12 ⊢ (𝑒 = ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) → ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(𝑒‘𝑎))𝐹𝑏))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏)))) ↔ (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏))))))
242	236, 241	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → (𝑒 = ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(𝑒‘𝑎))𝐹𝑏))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏))))))
243	242	expr 458	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ (𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁)) → (𝑐 ≠ 𝑓 → (𝑒 = ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(𝑒‘𝑎))𝐹𝑏))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏)))))))
244	243	impd 412	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ (𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁)) → ((𝑐 ≠ 𝑓 ∧ 𝑒 = ((pmTrsp‘𝑁)‘{𝑐, 𝑓})) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(𝑒‘𝑎))𝐹𝑏))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏))))))
245	244	rexlimdvva 3212	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) → (∃𝑐 ∈ 𝑁 ∃𝑓 ∈ 𝑁 (𝑐 ≠ 𝑓 ∧ 𝑒 = ((pmTrsp‘𝑁)‘{𝑐, 𝑓})) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(𝑒‘𝑎))𝐹𝑏))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏))))))
246	108, 245	syl5 34	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) → (𝑒 ∈ ran (pmTrsp‘𝑁) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(𝑒‘𝑎))𝐹𝑏))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏))))))
247	246	3impia 1118	. . . . . 6 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁 ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(𝑒‘𝑎))𝐹𝑏))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏)))))
248	106, 247	syl3an2 1165	. . . . 5 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘(𝑒‘𝑎))𝐹𝑏))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏)))))
249	105, 248	eqtrd 2773	. . . 4 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎)𝐹𝑏))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏)))))
250	249	adantr 482	. . 3 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) ∧ (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (𝐷‘𝐹))) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎)𝐹𝑏))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏)))))
251		fveq2 6889	. . . 4 ⊢ ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (𝐷‘𝐹)) → ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏)))) = ((inv_g‘𝑅)‘((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (𝐷‘𝐹))))
252	251	adantl 483	. . 3 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) ∧ (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (𝐷‘𝐹))) → ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏)))) = ((inv_g‘𝑅)‘((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (𝐷‘𝐹))))
253		eqid 2733	. . . . . 6 ⊢ (inv_g‘𝑅) = (inv_g‘𝑅)
254	47	3ad2ant1 1134	. . . . . 6 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑅 ∈ Ring)
255	58	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
256	255	3ad2ant1 1134	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
257	59, 52	mgpbas 19988	. . . . . . . . 9 ⊢ 𝐾 = (Base‘(mulGrp‘𝑅))
258	31, 257	mhmf 18674	. . . . . . . 8 ⊢ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾)
259	256, 258	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾)
260	259, 88	ffvelcdmd 7085	. . . . . 6 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) ∈ 𝐾)
261	49	3ad2ant1 1134	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝐷:𝐵⟶𝐾)
262		simp13 1206	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝐹 ∈ 𝐵)
263	261, 262	ffvelcdmd 7085	. . . . . 6 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → (𝐷‘𝐹) ∈ 𝐾)
264	52, 53, 253, 254, 260, 263	ringmneg1 20110	. . . . 5 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → (((inv_g‘𝑅)‘(((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑)) · (𝐷‘𝐹)) = ((inv_g‘𝑅)‘((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (𝐷‘𝐹))))
265	59, 53	mgpplusg 19986	. . . . . . . . 9 ⊢ · = (+_g‘(mulGrp‘𝑅))
266	31, 30, 265	mhmlin 18676	. . . . . . . 8 ⊢ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ (Base‘(SymGrp‘𝑁))) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(𝑑(+_g‘(SymGrp‘𝑁))𝑒)) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑒)))
267	256, 88, 90, 266	syl3anc 1372	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(𝑑(+_g‘(SymGrp‘𝑁))𝑒)) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑒)))
268	33	3ad2ant1 1134	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑁 ∈ Fin)
269		simp3 1139	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑒 ∈ ran (pmTrsp‘𝑁))
270	34, 31, 38	pmtrodpm 21142	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑒 ∈ ((Base‘(SymGrp‘𝑁)) ∖ (pmEven‘𝑁)))
271	268, 269, 270	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑒 ∈ ((Base‘(SymGrp‘𝑁)) ∖ (pmEven‘𝑁)))
272		eqid 2733	. . . . . . . . . 10 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
273		eqid 2733	. . . . . . . . . 10 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁)
274	272, 273, 54, 31, 253	zrhpsgnodpm 21137	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑒 ∈ ((Base‘(SymGrp‘𝑁)) ∖ (pmEven‘𝑁))) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑒) = ((inv_g‘𝑅)‘ 1 ))
275	254, 268, 271, 274	syl3anc 1372	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑒) = ((inv_g‘𝑅)‘ 1 ))
276	275	oveq2d 7422	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑒)) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · ((inv_g‘𝑅)‘ 1 )))
277	52, 53, 54, 253, 254, 260	ringnegr 20109	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · ((inv_g‘𝑅)‘ 1 )) = ((inv_g‘𝑅)‘(((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑)))
278	267, 276, 277	3eqtrrd 2778	. . . . . 6 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → ((inv_g‘𝑅)‘(((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑)) = (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(𝑑(+_g‘(SymGrp‘𝑁))𝑒)))
279	278	oveq1d 7421	. . . . 5 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → (((inv_g‘𝑅)‘(((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑)) · (𝐷‘𝐹)) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(𝑑(+_g‘(SymGrp‘𝑁))𝑒)) · (𝐷‘𝐹)))
280	264, 279	eqtr3d 2775	. . . 4 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → ((inv_g‘𝑅)‘((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (𝐷‘𝐹))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(𝑑(+_g‘(SymGrp‘𝑁))𝑒)) · (𝐷‘𝐹)))
281	280	adantr 482	. . 3 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) ∧ (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (𝐷‘𝐹))) → ((inv_g‘𝑅)‘((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (𝐷‘𝐹))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(𝑑(+_g‘(SymGrp‘𝑁))𝑒)) · (𝐷‘𝐹)))
282	250, 252, 281	3eqtrd 2777	. 2 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) ∧ (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (𝐷‘𝐹))) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(𝑑(+_g‘(SymGrp‘𝑁))𝑒)) · (𝐷‘𝐹)))
283		simp2 1138	. . 3 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐸:𝑁–1-1-onto→𝑁)
284	34, 31	elsymgbas 19236	. . . 4 ⊢ (𝑁 ∈ Fin → (𝐸 ∈ (Base‘(SymGrp‘𝑁)) ↔ 𝐸:𝑁–1-1-onto→𝑁))
285	33, 284	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (𝐸 ∈ (Base‘(SymGrp‘𝑁)) ↔ 𝐸:𝑁–1-1-onto→𝑁))
286	283, 285	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐸 ∈ (Base‘(SymGrp‘𝑁)))
287	7, 14, 21, 28, 29, 30, 31, 37, 40, 45, 87, 282, 286	mndind 18706	1 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘𝐹)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ∖ cdif 3945 ⊆ wss 3948 ifcif 4528 {csn 4628 {cpr 4630 class class class wbr 5148 I cid 5573 × cxp 5674 dom cdm 5676 ran crn 5677 ↾ cres 5678 ∘ ccom 5680 Fn wfn 6536 ⟶wf 6537 –1-1-onto→wf1o 6540 ‘cfv 6541 (class class class)co 7406 ∈ cmpo 7408 ∘_f cof 7665 2_oc2o 8457 ↑_m cmap 8817 ≈ cen 8933 Fincfn 8936 Basecbs 17141 +_gcplusg 17194 ._rcmulr 17195 0_gc0g 17382 mrClscmrc 17524 Mndcmnd 18622 MndHom cmhm 18666 SubMndcsubmnd 18667 Grpcgrp 18816 inv_gcminusg 18817 SymGrpcsymg 19229 pmTrspcpmtr 19304 pmSgncpsgn 19352 pmEvencevpm 19353 mulGrpcmgp 19982 1_rcur 19999 Ringcrg 20050 ℤRHomczrh 21041 Mat cmat 21899

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-addf 11186 ax-mulf 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-sup 9434 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-xnn0 12542 df-z 12556 df-dec 12675 df-uz 12820 df-rp 12972 df-fz 13482 df-fzo 13625 df-seq 13964 df-exp 14025 df-hash 14288 df-word 14462 df-lsw 14510 df-concat 14518 df-s1 14543 df-substr 14588 df-pfx 14618 df-splice 14697 df-reverse 14706 df-s2 14796 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-hom 17218 df-cco 17219 df-0g 17384 df-gsum 17385 df-prds 17390 df-pws 17392 df-mre 17527 df-mrc 17528 df-acs 17530 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-mhm 18668 df-submnd 18669 df-efmnd 18747 df-grp 18819 df-minusg 18820 df-mulg 18946 df-subg 18998 df-ghm 19085 df-gim 19128 df-oppg 19205 df-symg 19230 df-pmtr 19305 df-psgn 19354 df-evpm 19355 df-cmn 19645 df-abl 19646 df-mgp 19983 df-ur 20000 df-ring 20052 df-cring 20053 df-oppr 20143 df-dvdsr 20164 df-unit 20165 df-invr 20195 df-dvr 20208 df-rnghom 20244 df-drng 20310 df-subrg 20354 df-sra 20778 df-rgmod 20779 df-cnfld 20938 df-zring 21011 df-zrh 21045 df-dsmm 21279 df-frlm 21294 df-mat 21900

This theorem is referenced by: mdetunilem8 22113

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