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Theorem mdetunilem7 22743

Description: Lemma for mdetuni 22747. (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 6881	. . . . . 6 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑎) = (𝑑‘𝑎))
2	1	oveq1d 7426	. . . . 5 ⊢ (𝑐 = 𝑑 → ((𝑐‘𝑎)𝐹𝑏) = ((𝑑‘𝑎)𝐹𝑏))
3	2	mpoeq3dv 7490	. . . 4 ⊢ (𝑐 = 𝑑 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏)))
4	3	fveq2d 6886	. . 3 ⊢ (𝑐 = 𝑑 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏))))
5		fveq2 6882	. . . 4 ⊢ (𝑐 = 𝑑 → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) = (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑))
6	5	oveq1d 7426	. . 3 ⊢ (𝑐 = 𝑑 → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) · (𝐷‘𝐹)) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (𝐷‘𝐹)))
7	4, 6	eqeq12d 2785	. 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 6881	. . . . . 6 ⊢ (𝑐 = (𝑑(+_g‘(SymGrp‘𝑁))𝑒) → (𝑐‘𝑎) = ((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎))
9	8	oveq1d 7426	. . . . 5 ⊢ (𝑐 = (𝑑(+_g‘(SymGrp‘𝑁))𝑒) → ((𝑐‘𝑎)𝐹𝑏) = (((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎)𝐹𝑏))
10	9	mpoeq3dv 7490	. . . 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 6886	. . 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 6882	. . . 4 ⊢ (𝑐 = (𝑑(+_g‘(SymGrp‘𝑁))𝑒) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) = (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(𝑑(+_g‘(SymGrp‘𝑁))𝑒)))
13	12	oveq1d 7426	. . 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 2785	. 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 6881	. . . . . 6 ⊢ (𝑐 = (0_g‘(SymGrp‘𝑁)) → (𝑐‘𝑎) = ((0_g‘(SymGrp‘𝑁))‘𝑎))
16	15	oveq1d 7426	. . . . 5 ⊢ (𝑐 = (0_g‘(SymGrp‘𝑁)) → ((𝑐‘𝑎)𝐹𝑏) = (((0_g‘(SymGrp‘𝑁))‘𝑎)𝐹𝑏))
17	16	mpoeq3dv 7490	. . . 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 6886	. . 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 6882	. . . 4 ⊢ (𝑐 = (0_g‘(SymGrp‘𝑁)) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) = (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(0_g‘(SymGrp‘𝑁))))
20	19	oveq1d 7426	. . 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 2785	. 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 6881	. . . . . 6 ⊢ (𝑐 = 𝐸 → (𝑐‘𝑎) = (𝐸‘𝑎))
23	22	oveq1d 7426	. . . . 5 ⊢ (𝑐 = 𝐸 → ((𝑐‘𝑎)𝐹𝑏) = ((𝐸‘𝑎)𝐹𝑏))
24	23	mpoeq3dv 7490	. . . 4 ⊢ (𝑐 = 𝐸 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)𝐹𝑏)))
25	24	fveq2d 6886	. . 3 ⊢ (𝑐 = 𝐸 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)𝐹𝑏))))
26		fveq2 6882	. . . 4 ⊢ (𝑐 = 𝐸 → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) = (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸))
27	26	oveq1d 7426	. . 3 ⊢ (𝑐 = 𝐸 → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) · (𝐷‘𝐹)) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘𝐹)))
28	25, 27	eqeq12d 2785	. 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 2769	. 2 ⊢ (0_g‘(SymGrp‘𝑁)) = (0_g‘(SymGrp‘𝑁))
30		eqid 2769	. 2 ⊢ (+_g‘(SymGrp‘𝑁)) = (+_g‘(SymGrp‘𝑁))
31		eqid 2769	. 2 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
32		mdetuni.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ Fin)
33	32	3ad2ant1 1149	. . 3 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝑁 ∈ Fin)
34		eqid 2769	. . . 4 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
35	34	symggrp 19469	. . 3 ⊢ (𝑁 ∈ Fin → (SymGrp‘𝑁) ∈ Grp)
36		grpmnd 19006	. . 3 ⊢ ((SymGrp‘𝑁) ∈ Grp → (SymGrp‘𝑁) ∈ Mnd)
37	33, 35, 36	3syl 19	. 2 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (SymGrp‘𝑁) ∈ Mnd)
38		eqid 2769	. . . 4 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁)
39	38, 34, 31	symgtrf 19538	. . 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 2769	. . . . . 6 ⊢ (mrCls‘(SubMnd‘(SymGrp‘𝑁))) = (mrCls‘(SubMnd‘(SymGrp‘𝑁)))
42	38, 34, 31, 41	symggen2 19540	. . . . 5 ⊢ (𝑁 ∈ Fin → ((mrCls‘(SubMnd‘(SymGrp‘𝑁)))‘ran (pmTrsp‘𝑁)) = (Base‘(SymGrp‘𝑁)))
43	32, 42	syl 18	. . . 4 ⊢ (𝜑 → ((mrCls‘(SubMnd‘(SymGrp‘𝑁)))‘ran (pmTrsp‘𝑁)) = (Base‘(SymGrp‘𝑁)))
44	43	eqcomd 2775	. . 3 ⊢ (𝜑 → (Base‘(SymGrp‘𝑁)) = ((mrCls‘(SubMnd‘(SymGrp‘𝑁)))‘ran (pmTrsp‘𝑁)))
45	44	3ad2ant1 1149	. 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 1149	. . . 4 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝑅 ∈ Ring)
48		mdetuni.ff	. . . . . 6 ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)
49	48	3ad2ant1 1149	. . . . 5 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐷:𝐵⟶𝐾)
50		simp3 1154	. . . . 5 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐹 ∈ 𝐵)
51	49, 50	ffvelcdmd 7081	. . . 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 20351	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐷‘𝐹) ∈ 𝐾) → ( 1 · (𝐷‘𝐹)) = (𝐷‘𝐹))
56	47, 51, 55	syl2anc 595	. . 3 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → ( 1 · (𝐷‘𝐹)) = (𝐷‘𝐹))
57		zrhpsgnmhm 21702	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
58	46, 32, 57	syl2anc 595	. . . . . 6 ⊢ (𝜑 → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
59		eqid 2769	. . . . . . . 8 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
60	59, 54	ringidval 20264	. . . . . . 7 ⊢ 1 = (0_g‘(mulGrp‘𝑅))
61	29, 60	mhm0 18851	. . . . . 6 ⊢ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(0_g‘(SymGrp‘𝑁))) = 1 )
62	58, 61	syl 18	. . . . 5 ⊢ (𝜑 → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(0_g‘(SymGrp‘𝑁))) = 1 )
63	62	3ad2ant1 1149	. . . 4 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(0_g‘(SymGrp‘𝑁))) = 1 )
64	63	oveq1d 7426	. . 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 19470	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ Fin → ( I ↾ 𝑁) = (0_g‘(SymGrp‘𝑁)))
66	32, 65	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ( I ↾ 𝑁) = (0_g‘(SymGrp‘𝑁)))
67	66	3ad2ant1 1149	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → ( I ↾ 𝑁) = (0_g‘(SymGrp‘𝑁)))
68	67	3ad2ant1 1149	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ( I ↾ 𝑁) = (0_g‘(SymGrp‘𝑁)))
69	68	fveq1d 6884	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (( I ↾ 𝑁)‘𝑎) = ((0_g‘(SymGrp‘𝑁))‘𝑎))
70		simp2 1153	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑎 ∈ 𝑁)
71		fvresi 7172	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝑁 → (( I ↾ 𝑁)‘𝑎) = 𝑎)
72	70, 71	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (( I ↾ 𝑁)‘𝑎) = 𝑎)
73	69, 72	eqtr3d 2806	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((0_g‘(SymGrp‘𝑁))‘𝑎) = 𝑎)
74	73	oveq1d 7426	. . . . . 6 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (((0_g‘(SymGrp‘𝑁))‘𝑎)𝐹𝑏) = (𝑎𝐹𝑏))
75	74	mpoeq3dva 7488	. . . . 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 22547	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
79	78	3ad2ant3 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐹 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
80		elmapi 8845	. . . . . . 7 ⊢ (𝐹 ∈ (𝐾 ↑_m (𝑁 × 𝑁)) → 𝐹:(𝑁 × 𝑁)⟶𝐾)
81		ffn 6706	. . . . . . 7 ⊢ (𝐹:(𝑁 × 𝑁)⟶𝐾 → 𝐹 Fn (𝑁 × 𝑁))
82	79, 80, 81	3syl 19	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐹 Fn (𝑁 × 𝑁))
83		fnov 7542	. . . . . 6 ⊢ (𝐹 Fn (𝑁 × 𝑁) ↔ 𝐹 = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (𝑎𝐹𝑏)))
84	82, 83	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐹 = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (𝑎𝐹𝑏)))
85	75, 84	eqtr4d 2807	. . . 4 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((0_g‘(SymGrp‘𝑁))‘𝑎)𝐹𝑏)) = 𝐹)
86	85	fveq2d 6886	. . 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 2815	. 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 1153	. . . . . . . . . . . 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 3941	. . . . . . . . . . . . 13 ⊢ (𝑒 ∈ ran (pmTrsp‘𝑁) → 𝑒 ∈ (Base‘(SymGrp‘𝑁)))
90	89	3ad2ant3 1151	. . . . . . . . . . . 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 19453	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ (Base‘(SymGrp‘𝑁))) → (𝑑(+_g‘(SymGrp‘𝑁))𝑒) = (𝑑 ∘ 𝑒))
92	88, 90, 91	syl2anc 595	. . . . . . . . . . 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 6884	. . . . . . . . . 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 1149	. . . . . . . . 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 19444	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ (Base‘(SymGrp‘𝑁)) → 𝑒:𝑁–1-1-onto→𝑁)
96		f1of 6821	. . . . . . . . . . . 12 ⊢ (𝑒:𝑁–1-1-onto→𝑁 → 𝑒:𝑁⟶𝑁)
97	90, 95, 96	3syl 19	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑒:𝑁⟶𝑁)
98	97	3ad2ant1 1149	. . . . . . . . . 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 1153	. . . . . . . . . 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 6982	. . . . . . . . . 10 ⊢ ((𝑒:𝑁⟶𝑁 ∧ 𝑎 ∈ 𝑁) → ((𝑑 ∘ 𝑒)‘𝑎) = (𝑑‘(𝑒‘𝑎)))
101	98, 99, 100	syl2anc 595	. . . . . . . . 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 2804	. . . . . . . 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 7426	. . . . . . 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 7488	. . . . . 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 6886	. . . . 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 19445	. . . . . 6 ⊢ (𝑑 ∈ (Base‘(SymGrp‘𝑁)) → 𝑑:𝑁⟶𝑁)
107		eqid 2769	. . . . . . . . 9 ⊢ (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
108	107, 38	pmtrrn2 19529	. . . . . . . 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 1229	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → 𝜑)
115		df-3an 1103	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁 ∧ 𝑐 ≠ 𝑓) ↔ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓))
116	115	bilanri 511	. . . . . . . . . . . . . 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))
117	79, 80	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐹:(𝑁 × 𝑁)⟶𝐾)
118	117	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) → 𝐹:(𝑁 × 𝑁)⟶𝐾)
119	118	ad2antrr 738	. . . . . . . . . . . . . . . 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)
120		simpllr 787	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑏 ∈ 𝑁) → 𝑑:𝑁⟶𝑁)
121		simprlr 791	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → 𝑓 ∈ 𝑁)
122	121	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑏 ∈ 𝑁) → 𝑓 ∈ 𝑁)
123	120, 122	ffvelcdmd 7081	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑏 ∈ 𝑁) → (𝑑‘𝑓) ∈ 𝑁)
124		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑏 ∈ 𝑁) → 𝑏 ∈ 𝑁)
125	119, 123, 124	fovcdmd 7583	. . . . . . . . . . . . . . 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)
126		simprll 790	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → 𝑐 ∈ 𝑁)
127	126	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑏 ∈ 𝑁) → 𝑐 ∈ 𝑁)
128	120, 127	ffvelcdmd 7081	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑏 ∈ 𝑁) → (𝑑‘𝑐) ∈ 𝑁)
129	119, 128, 124	fovcdmd 7583	. . . . . . . . . . . . . . 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)
130	125, 129	jca 520	. . . . . . . . . . . . . 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))
131	117	ad2antrr 738	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → 𝐹:(𝑁 × 𝑁)⟶𝐾)
132	131	3ad2ant1 1149	. . . . . . . . . . . . . . 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)
133		simp1lr 1254	. . . . . . . . . . . . . . . 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)
134		simp2 1153	. . . . . . . . . . . . . . . 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)
135	133, 134	ffvelcdmd 7081	. . . . . . . . . . . . . . 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)
136		simp3 1154	. . . . . . . . . . . . . . 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)
137	132, 135, 136	fovcdmd 7583	. . . . . . . . . . . . . 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)
138	76, 77, 52, 109, 54, 110, 53, 32, 46, 48, 111, 112, 113, 114, 116, 130, 137	mdetunilem6 22742	. . . . . . . . . . . . 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)))))))
139		simpl1 1208	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) → 𝜑)
140		fveq2 6882	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑐 → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑐))
141	32	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → 𝑁 ∈ Fin)
142		simprll 790	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → 𝑐 ∈ 𝑁)
143		simprlr 791	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → 𝑓 ∈ 𝑁)
144		simprr 784	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → 𝑐 ≠ 𝑓)
145	107	pmtrprfv 19522	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ Fin ∧ (𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁 ∧ 𝑐 ≠ 𝑓)) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑐) = 𝑓)
146	141, 142, 143, 144, 145	syl13anc 1397	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑐) = 𝑓)
147	146	adantr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑐) = 𝑓)
148	140, 147	sylan9eqr 2826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑐) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = 𝑓)
149	148	fveq2d 6886	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑐) → (𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎)) = (𝑑‘𝑓))
150	149	oveq1d 7426	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑐) → ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏) = ((𝑑‘𝑓)𝐹𝑏))
151		iftrue 4498	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑐 → if(𝑎 = 𝑐, ((𝑑‘𝑓)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))) = ((𝑑‘𝑓)𝐹𝑏))
152	151	adantl 486	. . . . . . . . . . . . . . . . . . 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))
153	150, 152	eqtr4d 2807	. . . . . . . . . . . . . . . . . 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))))
154		fveq2 6882	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑓 → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑓))
155		prcom 4703	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ {𝑐, 𝑓} = {𝑓, 𝑐}
156	155	fveq2i 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) = ((pmTrsp‘𝑁)‘{𝑓, 𝑐})
157	156	fveq1i 6883	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑓) = (((pmTrsp‘𝑁)‘{𝑓, 𝑐})‘𝑓)
158	32	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ Fin)
159		simplrl 788	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → (𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁))
160	159	simprd 500	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → 𝑓 ∈ 𝑁)
161	159	simpld 499	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → 𝑐 ∈ 𝑁)
162		simplrr 789	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → 𝑐 ≠ 𝑓)
163	162	necomd 3019	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → 𝑓 ≠ 𝑐)
164	107	pmtrprfv 19522	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑁 ∈ Fin ∧ (𝑓 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁 ∧ 𝑓 ≠ 𝑐)) → (((pmTrsp‘𝑁)‘{𝑓, 𝑐})‘𝑓) = 𝑐)
165	158, 160, 161, 163, 164	syl13anc 1397	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → (((pmTrsp‘𝑁)‘{𝑓, 𝑐})‘𝑓) = 𝑐)
166	157, 165	eqtrid 2816	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑓) = 𝑐)
167	154, 166	sylan9eqr 2826	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑓) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = 𝑐)
168	167	fveq2d 6886	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑓) → (𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎)) = (𝑑‘𝑐))
169	168	oveq1d 7426	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑓) → ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏) = ((𝑑‘𝑐)𝐹𝑏))
170		iftrue 4498	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑓 → if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)) = ((𝑑‘𝑐)𝐹𝑏))
171	170	adantl 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑓) → if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)) = ((𝑑‘𝑐)𝐹𝑏))
172	169, 171	eqtr4d 2807	. . . . . . . . . . . . . . . . . . . . 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)))
173	172	adantlr 727	. . . . . . . . . . . . . . . . . . . 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)))
174		vex 3467	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝑎 ∈ V
175	174	elpr 4619	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 ∈ {𝑐, 𝑓} ↔ (𝑎 = 𝑐 ∨ 𝑎 = 𝑓))
176	175	notbii 323	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (¬ 𝑎 ∈ {𝑐, 𝑓} ↔ ¬ (𝑎 = 𝑐 ∨ 𝑎 = 𝑓))
177		ioran 999	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (¬ (𝑎 = 𝑐 ∨ 𝑎 = 𝑓) ↔ (¬ 𝑎 = 𝑐 ∧ ¬ 𝑎 = 𝑓))
178	176, 177	sylbbr 239	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((¬ 𝑎 = 𝑐 ∧ ¬ 𝑎 = 𝑓) → ¬ 𝑎 ∈ {𝑐, 𝑓})
179	178	adantll 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ ¬ 𝑎 = 𝑓) → ¬ 𝑎 ∈ {𝑐, 𝑓})
180		prssi 4791	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) → {𝑐, 𝑓} ⊆ 𝑁)
181	159, 180	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → {𝑐, 𝑓} ⊆ 𝑁)
182		pr2ne 9988	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) → ({𝑐, 𝑓} ≈ 2_o ↔ 𝑐 ≠ 𝑓))
183	159, 182	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → ({𝑐, 𝑓} ≈ 2_o ↔ 𝑐 ≠ 𝑓))
184	162, 183	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → {𝑐, 𝑓} ≈ 2_o)
185	107	pmtrmvd 19525	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ Fin ∧ {𝑐, 𝑓} ⊆ 𝑁 ∧ {𝑐, 𝑓} ≈ 2_o) → dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I ) = {𝑐, 𝑓})
186	158, 181, 184, 185	syl3anc 1396	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I ) = {𝑐, 𝑓})
187	186	eleq2d 2855	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → (𝑎 ∈ dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I ) ↔ 𝑎 ∈ {𝑐, 𝑓}))
188	187	notbid 321	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → (¬ 𝑎 ∈ dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I ) ↔ ¬ 𝑎 ∈ {𝑐, 𝑓}))
189	188	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ ¬ 𝑎 = 𝑓) → (¬ 𝑎 ∈ dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I ) ↔ ¬ 𝑎 ∈ {𝑐, 𝑓}))
190	179, 189	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ ¬ 𝑎 = 𝑓) → ¬ 𝑎 ∈ dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I ))
191	107	pmtrf 19524	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 ∈ Fin ∧ {𝑐, 𝑓} ⊆ 𝑁 ∧ {𝑐, 𝑓} ≈ 2_o) → ((pmTrsp‘𝑁)‘{𝑐, 𝑓}):𝑁⟶𝑁)
192	158, 181, 184, 191	syl3anc 1396	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → ((pmTrsp‘𝑁)‘{𝑐, 𝑓}):𝑁⟶𝑁)
193	192	ffnd 6707	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) Fn 𝑁)
194		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → 𝑎 ∈ 𝑁)
195		fnelnfp 7176	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((pmTrsp‘𝑁)‘{𝑐, 𝑓}) Fn 𝑁 ∧ 𝑎 ∈ 𝑁) → (𝑎 ∈ dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I ) ↔ (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) ≠ 𝑎))
196	195	necon2bbid 3007	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((pmTrsp‘𝑁)‘{𝑐, 𝑓}) Fn 𝑁 ∧ 𝑎 ∈ 𝑁) → ((((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = 𝑎 ↔ ¬ 𝑎 ∈ dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I )))
197	193, 194, 196	syl2anc 595	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → ((((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = 𝑎 ↔ ¬ 𝑎 ∈ dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I )))
198	197	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . 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 )))
199	190, 198	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ ¬ 𝑎 = 𝑓) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = 𝑎)
200	199	fveq2d 6886	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ ¬ 𝑎 = 𝑓) → (𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎)) = (𝑑‘𝑎))
201	200	oveq1d 7426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ ¬ 𝑎 = 𝑓) → ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏) = ((𝑑‘𝑎)𝐹𝑏))
202		iffalse 4501	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑎 = 𝑓 → if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)) = ((𝑑‘𝑎)𝐹𝑏))
203	202	adantl 486	. . . . . . . . . . . . . . . . . . . . 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))
204	201, 203	eqtr4d 2807	. . . . . . . . . . . . . . . . . . . 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)))
205	173, 204	pm2.61dan 824	. . . . . . . . . . . . . . . . . . 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)))
206		iffalse 4501	. . . . . . . . . . . . . . . . . . . 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)))
207	206	adantl 486	. . . . . . . . . . . . . . . . . . 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)))
208	205, 207	eqtr4d 2807	. . . . . . . . . . . . . . . . . 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))))
209	153, 208	pm2.61dan 824	. . . . . . . . . . . . . . . . 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))))
210	209	3adant3 1148	. . . . . . . . . . . . . . . 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))))
211	210	mpoeq3dva 7488	. . . . . . . . . . . . . . 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)))))
212	139, 211	sylan 591	. . . . . . . . . . . . . 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)))))
213	212	fveq2d 6886	. . . . . . . . . . . . 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))))))
214		fveq2 6882	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑐 → (𝑑‘𝑎) = (𝑑‘𝑐))
215	214	oveq1d 7426	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑐 → ((𝑑‘𝑎)𝐹𝑏) = ((𝑑‘𝑐)𝐹𝑏))
216		iftrue 4498	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑐 → if(𝑎 = 𝑐, ((𝑑‘𝑐)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))) = ((𝑑‘𝑐)𝐹𝑏))
217	215, 216	eqtr4d 2807	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑐 → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑐, ((𝑑‘𝑐)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))))
218		fveq2 6882	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑓 → (𝑑‘𝑎) = (𝑑‘𝑓))
219	218	oveq1d 7426	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑓 → ((𝑑‘𝑎)𝐹𝑏) = ((𝑑‘𝑓)𝐹𝑏))
220		iftrue 4498	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑓 → if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)) = ((𝑑‘𝑓)𝐹𝑏))
221	219, 220	eqtr4d 2807	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑓 → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
222	221	adantl 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ 𝑎 = 𝑐 ∧ 𝑎 = 𝑓) → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
223		iffalse 4501	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝑎 = 𝑓 → if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)) = ((𝑑‘𝑎)𝐹𝑏))
224	223	eqcomd 2775	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑎 = 𝑓 → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
225	224	adantl 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ 𝑎 = 𝑐 ∧ ¬ 𝑎 = 𝑓) → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
226	222, 225	pm2.61dan 824	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑎 = 𝑐 → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
227		iffalse 4501	. . . . . . . . . . . . . . . . . . . 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)))
228	226, 227	eqtr4d 2807	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑎 = 𝑐 → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑐, ((𝑑‘𝑐)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))))
229	217, 228	pm2.61i 184	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑐, ((𝑑‘𝑐)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
230	229	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))))
231	230	mpoeq3ia 7489	. . . . . . . . . . . . . . . 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))))
232	231	fveq2i 6885	. . . . . . . . . . . . . . 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)))))
233	232	fveq2i 6885	. . . . . . . . . . . . . 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))))))
234	233	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)))))))
235	138, 213, 234	3eqtr4d 2814	. . . . . . . . . . . 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)))))
236		fveq1 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) → (𝑒‘𝑎) = (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))
237	236	fveq2d 6886	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) → (𝑑‘(𝑒‘𝑎)) = (𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎)))
238	237	oveq1d 7426	. . . . . . . . . . . . . 14 ⊢ (𝑒 = ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) → ((𝑑‘(𝑒‘𝑎))𝐹𝑏) = ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏))
239	238	mpoeq3dv 7490	. . . . . . . . . . . . 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)))
240	239	fveqeq2d 6890	. . . . . . . . . . . 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))))))
241	235, 240	syl5ibrcom 250	. . . . . . . . . . 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))))))
242	241	expr 461	. . . . . . . . . 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)))))))
243	242	impd 415	. . . . . . . . 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))))))
244	243	rexlimdvva 3228	. . . . . . . 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))))))
245	108, 244	syl5 35	. . . . . . 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))))))
246	245	3impia 1133	. . . . . 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)))))
247	106, 246	syl3an2 1180	. . . . 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)))))
248	105, 247	eqtrd 2804	. . . 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)))))
249	248	adantr 485	. . 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)))))
250		fveq2 6882	. . . 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))))
251	250	adantl 486	. . 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))))
252		eqid 2769	. . . . . 6 ⊢ (inv_g‘𝑅) = (inv_g‘𝑅)
253	47	3ad2ant1 1149	. . . . . 6 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑅 ∈ Ring)
254	58	3ad2ant1 1149	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
255	254	3ad2ant1 1149	. . . . . . . 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)))
256	59, 52	mgpbas 20220	. . . . . . . . 9 ⊢ 𝐾 = (Base‘(mulGrp‘𝑅))
257	31, 256	mhmf 18846	. . . . . . . 8 ⊢ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾)
258	255, 257	syl 18	. . . . . . 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)
259	258, 88	ffvelcdmd 7081	. . . . . 6 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) ∈ 𝐾)
260	49	3ad2ant1 1149	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝐷:𝐵⟶𝐾)
261		simp13 1222	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝐹 ∈ 𝐵)
262	260, 261	ffvelcdmd 7081	. . . . . 6 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → (𝐷‘𝐹) ∈ 𝐾)
263	52, 53, 252, 253, 259, 262	ringmneg1 20386	. . . . 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))))
264	59, 53	mgpplusg 20219	. . . . . . . . 9 ⊢ · = (+_g‘(mulGrp‘𝑅))
265	31, 30, 264	mhmlin 18850	. . . . . . . 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)))
266	255, 88, 90, 265	syl3anc 1396	. . . . . . 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)))
267	33	3ad2ant1 1149	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑁 ∈ Fin)
268		simp3 1154	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑒 ∈ ran (pmTrsp‘𝑁))
269	34, 31, 38	pmtrodpm 21715	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑒 ∈ ((Base‘(SymGrp‘𝑁)) ∖ (pmEven‘𝑁)))
270	267, 268, 269	syl2anc 595	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑒 ∈ ((Base‘(SymGrp‘𝑁)) ∖ (pmEven‘𝑁)))
271		eqid 2769	. . . . . . . . . 10 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
272		eqid 2769	. . . . . . . . . 10 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁)
273	271, 272, 54, 31, 252	zrhpsgnodpm 21710	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑒 ∈ ((Base‘(SymGrp‘𝑁)) ∖ (pmEven‘𝑁))) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑒) = ((inv_g‘𝑅)‘ 1 ))
274	253, 267, 270, 273	syl3anc 1396	. . . . . . . 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 ))
275	274	oveq2d 7427	. . . . . . 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 )))
276	52, 53, 54, 252, 253, 259	ringnegr 20385	. . . . . . 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)))
277	266, 275, 276	3eqtrrd 2809	. . . . . 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)))
278	277	oveq1d 7426	. . . . 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)))
279	263, 278	eqtr3d 2806	. . . 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)))
280	279	adantr 485	. . 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)))
281	249, 251, 280	3eqtrd 2808	. 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)))
282		simp2 1153	. . 3 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐸:𝑁–1-1-onto→𝑁)
283	34, 31	elsymgbas 19443	. . . 4 ⊢ (𝑁 ∈ Fin → (𝐸 ∈ (Base‘(SymGrp‘𝑁)) ↔ 𝐸:𝑁–1-1-onto→𝑁))
284	33, 283	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (𝐸 ∈ (Base‘(SymGrp‘𝑁)) ↔ 𝐸:𝑁–1-1-onto→𝑁))
285	282, 284	mpbird 260	. 2 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐸 ∈ (Base‘(SymGrp‘𝑁)))
286	7, 14, 21, 28, 29, 30, 31, 37, 40, 45, 87, 281, 285	mndind 18886	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 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 ∖ cdif 3910 ⊆ wss 3913 ifcif 4492 {csn 4594 {cpr 4596 class class class wbr 5113 I cid 5556 × cxp 5660 dom cdm 5662 ran crn 5663 ↾ cres 5664 ∘ ccom 5666 Fn wfn 6532 ⟶wf 6533 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7411 ∈ cmpo 7413 ∘_f cof 7673 2_oc2o 8446 ↑_m cmap 8823 ≈ cen 8939 Fincfn 8942 Basecbs 17268 +_gcplusg 17309 ._rcmulr 17310 0_gc0g 17491 mrClscmrc 17634 Mndcmnd 18791 MndHom cmhm 18838 SubMndcsubmnd 18839 Grpcgrp 18999 inv_gcminusg 19000 SymGrpcsymg 19438 pmTrspcpmtr 19510 pmSgncpsgn 19558 pmEvencevpm 19559 mulGrpcmgp 20215 1_rcur 20262 Ringcrg 20314 ℤRHomczrh 21617 Mat cmat 22532

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-addf 11178 ax-mulf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-xor 1539 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-ot 4603 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-tpos 8221 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-map 8825 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-sup 9401 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-xnn0 12577 df-z 12591 df-dec 12711 df-uz 12862 df-rp 13016 df-fz 13535 df-fzo 13682 df-seq 14037 df-exp 14097 df-hash 14366 df-word 14550 df-lsw 14599 df-concat 14607 df-s1 14633 df-substr 14678 df-pfx 14708 df-splice 14786 df-reverse 14795 df-s2 14884 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-hom 17333 df-cco 17334 df-0g 17493 df-gsum 17494 df-prds 17499 df-pws 17501 df-mre 17637 df-mrc 17638 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-mhm 18840 df-submnd 18841 df-efmnd 18927 df-grp 19002 df-minusg 19003 df-mulg 19133 df-subg 19188 df-ghm 19283 df-gim 19328 df-oppg 19415 df-symg 19439 df-pmtr 19511 df-psgn 19560 df-evpm 19561 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-cring 20317 df-oppr 20418 df-dvdsr 20438 df-unit 20439 df-invr 20469 df-dvr 20482 df-rhm 20553 df-subrng 20630 df-subrg 20654 df-drng 20814 df-sra 21271 df-rgmod 21272 df-cnfld 21491 df-zring 21565 df-zrh 21621 df-dsmm 21850 df-frlm 21865 df-mat 22533

This theorem is referenced by: mdetunilem8 22744

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