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Theorem mdetunilem7 22560

Description: Lemma for mdetuni 22564. (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 6831	. . . . . 6 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑎) = (𝑑‘𝑎))
2	1	oveq1d 7371	. . . . 5 ⊢ (𝑐 = 𝑑 → ((𝑐‘𝑎)𝐹𝑏) = ((𝑑‘𝑎)𝐹𝑏))
3	2	mpoeq3dv 7435	. . . 4 ⊢ (𝑐 = 𝑑 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏)))
4	3	fveq2d 6836	. . 3 ⊢ (𝑐 = 𝑑 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑑‘𝑎)𝐹𝑏))))
5		fveq2 6832	. . . 4 ⊢ (𝑐 = 𝑑 → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) = (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑))
6	5	oveq1d 7371	. . 3 ⊢ (𝑐 = 𝑑 → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) · (𝐷‘𝐹)) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑑) · (𝐷‘𝐹)))
7	4, 6	eqeq12d 2750	. 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 6831	. . . . . 6 ⊢ (𝑐 = (𝑑(+_g‘(SymGrp‘𝑁))𝑒) → (𝑐‘𝑎) = ((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎))
9	8	oveq1d 7371	. . . . 5 ⊢ (𝑐 = (𝑑(+_g‘(SymGrp‘𝑁))𝑒) → ((𝑐‘𝑎)𝐹𝑏) = (((𝑑(+_g‘(SymGrp‘𝑁))𝑒)‘𝑎)𝐹𝑏))
10	9	mpoeq3dv 7435	. . . 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 6836	. . 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 6832	. . . 4 ⊢ (𝑐 = (𝑑(+_g‘(SymGrp‘𝑁))𝑒) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) = (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(𝑑(+_g‘(SymGrp‘𝑁))𝑒)))
13	12	oveq1d 7371	. . 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 2750	. 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 6831	. . . . . 6 ⊢ (𝑐 = (0_g‘(SymGrp‘𝑁)) → (𝑐‘𝑎) = ((0_g‘(SymGrp‘𝑁))‘𝑎))
16	15	oveq1d 7371	. . . . 5 ⊢ (𝑐 = (0_g‘(SymGrp‘𝑁)) → ((𝑐‘𝑎)𝐹𝑏) = (((0_g‘(SymGrp‘𝑁))‘𝑎)𝐹𝑏))
17	16	mpoeq3dv 7435	. . . 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 6836	. . 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 6832	. . . 4 ⊢ (𝑐 = (0_g‘(SymGrp‘𝑁)) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) = (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(0_g‘(SymGrp‘𝑁))))
20	19	oveq1d 7371	. . 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 2750	. 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 6831	. . . . . 6 ⊢ (𝑐 = 𝐸 → (𝑐‘𝑎) = (𝐸‘𝑎))
23	22	oveq1d 7371	. . . . 5 ⊢ (𝑐 = 𝐸 → ((𝑐‘𝑎)𝐹𝑏) = ((𝐸‘𝑎)𝐹𝑏))
24	23	mpoeq3dv 7435	. . . 4 ⊢ (𝑐 = 𝐸 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)𝐹𝑏)))
25	24	fveq2d 6836	. . 3 ⊢ (𝑐 = 𝐸 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝑐‘𝑎)𝐹𝑏))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)𝐹𝑏))))
26		fveq2 6832	. . . 4 ⊢ (𝑐 = 𝐸 → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) = (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸))
27	26	oveq1d 7371	. . 3 ⊢ (𝑐 = 𝐸 → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑐) · (𝐷‘𝐹)) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘𝐹)))
28	25, 27	eqeq12d 2750	. 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 2734	. 2 ⊢ (0_g‘(SymGrp‘𝑁)) = (0_g‘(SymGrp‘𝑁))
30		eqid 2734	. 2 ⊢ (+_g‘(SymGrp‘𝑁)) = (+_g‘(SymGrp‘𝑁))
31		eqid 2734	. 2 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
32		mdetuni.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ Fin)
33	32	3ad2ant1 1133	. . 3 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝑁 ∈ Fin)
34		eqid 2734	. . . 4 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
35	34	symggrp 19327	. . 3 ⊢ (𝑁 ∈ Fin → (SymGrp‘𝑁) ∈ Grp)
36		grpmnd 18868	. . 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 2734	. . . 4 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁)
39	38, 34, 31	symgtrf 19396	. . 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 2734	. . . . . 6 ⊢ (mrCls‘(SubMnd‘(SymGrp‘𝑁))) = (mrCls‘(SubMnd‘(SymGrp‘𝑁)))
42	38, 34, 31, 41	symggen2 19398	. . . . 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 2740	. . 3 ⊢ (𝜑 → (Base‘(SymGrp‘𝑁)) = ((mrCls‘(SubMnd‘(SymGrp‘𝑁)))‘ran (pmTrsp‘𝑁)))
45	44	3ad2ant1 1133	. 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 1133	. . . 4 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝑅 ∈ Ring)
48		mdetuni.ff	. . . . . 6 ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)
49	48	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐷:𝐵⟶𝐾)
50		simp3 1138	. . . . 5 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐹 ∈ 𝐵)
51	49, 50	ffvelcdmd 7028	. . . 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 20202	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐷‘𝐹) ∈ 𝐾) → ( 1 · (𝐷‘𝐹)) = (𝐷‘𝐹))
56	47, 51, 55	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → ( 1 · (𝐷‘𝐹)) = (𝐷‘𝐹))
57		zrhpsgnmhm 21537	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
58	46, 32, 57	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
59		eqid 2734	. . . . . . . 8 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
60	59, 54	ringidval 20116	. . . . . . 7 ⊢ 1 = (0_g‘(mulGrp‘𝑅))
61	29, 60	mhm0 18717	. . . . . 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 1133	. . . 4 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘(0_g‘(SymGrp‘𝑁))) = 1 )
64	63	oveq1d 7371	. . 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 19328	. . . . . . . . . . . 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 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → ( I ↾ 𝑁) = (0_g‘(SymGrp‘𝑁)))
68	67	3ad2ant1 1133	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ( I ↾ 𝑁) = (0_g‘(SymGrp‘𝑁)))
69	68	fveq1d 6834	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (( I ↾ 𝑁)‘𝑎) = ((0_g‘(SymGrp‘𝑁))‘𝑎))
70		simp2 1137	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑎 ∈ 𝑁)
71		fvresi 7117	. . . . . . . . 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 2771	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((0_g‘(SymGrp‘𝑁))‘𝑎) = 𝑎)
74	73	oveq1d 7371	. . . . . 6 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (((0_g‘(SymGrp‘𝑁))‘𝑎)𝐹𝑏) = (𝑎𝐹𝑏))
75	74	mpoeq3dva 7433	. . . . 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 22364	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
79	78	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐹 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
80		elmapi 8784	. . . . . . 7 ⊢ (𝐹 ∈ (𝐾 ↑_m (𝑁 × 𝑁)) → 𝐹:(𝑁 × 𝑁)⟶𝐾)
81		ffn 6660	. . . . . . 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 7487	. . . . . 6 ⊢ (𝐹 Fn (𝑁 × 𝑁) ↔ 𝐹 = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (𝑎𝐹𝑏)))
84	82, 83	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐹 = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (𝑎𝐹𝑏)))
85	75, 84	eqtr4d 2772	. . . 4 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (((0_g‘(SymGrp‘𝑁))‘𝑎)𝐹𝑏)) = 𝐹)
86	85	fveq2d 6836	. . 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 2780	. 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 1137	. . . . . . . . . . . 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 3927	. . . . . . . . . . . . 13 ⊢ (𝑒 ∈ ran (pmTrsp‘𝑁) → 𝑒 ∈ (Base‘(SymGrp‘𝑁)))
90	89	3ad2ant3 1135	. . . . . . . . . . . 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 19311	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ (Base‘(SymGrp‘𝑁))) → (𝑑(+_g‘(SymGrp‘𝑁))𝑒) = (𝑑 ∘ 𝑒))
92	88, 90, 91	syl2anc 584	. . . . . . . . . . 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 6834	. . . . . . . . . 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 1133	. . . . . . . . 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 19302	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ (Base‘(SymGrp‘𝑁)) → 𝑒:𝑁–1-1-onto→𝑁)
96		f1of 6772	. . . . . . . . . . . 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 1133	. . . . . . . . . 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 1137	. . . . . . . . . 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 6931	. . . . . . . . . 10 ⊢ ((𝑒:𝑁⟶𝑁 ∧ 𝑎 ∈ 𝑁) → ((𝑑 ∘ 𝑒)‘𝑎) = (𝑑‘(𝑒‘𝑎)))
101	98, 99, 100	syl2anc 584	. . . . . . . . 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 2769	. . . . . . . 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 7371	. . . . . . 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 7433	. . . . . 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 6836	. . . . 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 19303	. . . . . 6 ⊢ (𝑑 ∈ (Base‘(SymGrp‘𝑁)) → 𝑑:𝑁⟶𝑁)
107		eqid 2734	. . . . . . . . 9 ⊢ (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
108	107, 38	pmtrrn2 19387	. . . . . . . 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 1088	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁 ∧ 𝑐 ≠ 𝑓) ↔ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓))
116	115	biimpri 228	. . . . . . . . . . . . . . 15 ⊢ (((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓) → (𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁 ∧ 𝑐 ≠ 𝑓))
117	116	adantl 481	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑:𝑁⟶𝑁) → 𝐹:(𝑁 × 𝑁)⟶𝐾)
120	119	ad2antrr 726	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 7028	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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 7528	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 7028	. . . . . . . . . . . . . . . 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 7528	. . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . . 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 7028	. . . . . . . . . . . . . . 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 1138	. . . . . . . . . . . . . . 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 7528	. . . . . . . . . . . . . 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 22559	. . . . . . . . . . . . 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 6832	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑐 → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑐))
142	32	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 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 19380	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ Fin ∧ (𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁 ∧ 𝑐 ≠ 𝑓)) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑐) = 𝑓)
147	142, 143, 144, 145, 146	syl13anc 1374	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑐) = 𝑓)
148	147	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑐) = 𝑓)
149	141, 148	sylan9eqr 2791	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑐) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = 𝑓)
150	149	fveq2d 6836	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑐) → (𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎)) = (𝑑‘𝑓))
151	150	oveq1d 7371	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑐) → ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏) = ((𝑑‘𝑓)𝐹𝑏))
152		iftrue 4483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑐 → if(𝑎 = 𝑐, ((𝑑‘𝑓)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))) = ((𝑑‘𝑓)𝐹𝑏))
153	152	adantl 481	. . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . 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 6832	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑓 → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑓))
156		prcom 4687	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ {𝑐, 𝑓} = {𝑓, 𝑐}
157	156	fveq2i 6835	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) = ((pmTrsp‘𝑁)‘{𝑓, 𝑐})
158	157	fveq1i 6833	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑓) = (((pmTrsp‘𝑁)‘{𝑓, 𝑐})‘𝑓)
159	32	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → 𝑓 ∈ 𝑁)
162	160	simpld 494	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2985	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → 𝑓 ≠ 𝑐)
165	107	pmtrprfv 19380	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑁 ∈ Fin ∧ (𝑓 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁 ∧ 𝑓 ≠ 𝑐)) → (((pmTrsp‘𝑁)‘{𝑓, 𝑐})‘𝑓) = 𝑐)
166	159, 161, 162, 164, 165	syl13anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → (((pmTrsp‘𝑁)‘{𝑓, 𝑐})‘𝑓) = 𝑐)
167	158, 166	eqtrid 2781	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑓) = 𝑐)
168	155, 167	sylan9eqr 2791	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑓) → (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎) = 𝑐)
169	168	fveq2d 6836	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑓) → (𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎)) = (𝑑‘𝑐))
170	169	oveq1d 7371	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ 𝑎 = 𝑓) → ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏) = ((𝑑‘𝑐)𝐹𝑏))
171		iftrue 4483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑓 → if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)) = ((𝑑‘𝑐)𝐹𝑏))
172	171	adantl 481	. . . . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . . . 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 3442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝑎 ∈ V
176	175	elpr 4603	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 ∈ {𝑐, 𝑓} ↔ (𝑎 = 𝑐 ∨ 𝑎 = 𝑓))
177	176	notbii 320	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (¬ 𝑎 ∈ {𝑐, 𝑓} ↔ ¬ (𝑎 = 𝑐 ∨ 𝑎 = 𝑓))
178		ioran 985	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (¬ (𝑎 = 𝑐 ∨ 𝑎 = 𝑓) ↔ (¬ 𝑎 = 𝑐 ∧ ¬ 𝑎 = 𝑓))
179	177, 178	sylbbr 236	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((¬ 𝑎 = 𝑐 ∧ ¬ 𝑎 = 𝑓) → ¬ 𝑎 ∈ {𝑐, 𝑓})
180	179	adantll 714	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑐) ∧ ¬ 𝑎 = 𝑓) → ¬ 𝑎 ∈ {𝑐, 𝑓})
181		prssi 4775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9913	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 19383	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ Fin ∧ {𝑐, 𝑓} ⊆ 𝑁 ∧ {𝑐, 𝑓} ≈ 2_o) → dom (((pmTrsp‘𝑁)‘{𝑐, 𝑓}) ∖ I ) = {𝑐, 𝑓})
187	159, 182, 185, 186	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . . . 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 19382	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 ∈ Fin ∧ {𝑐, 𝑓} ⊆ 𝑁 ∧ {𝑐, 𝑓} ≈ 2_o) → ((pmTrsp‘𝑁)‘{𝑐, 𝑓}):𝑁⟶𝑁)
193	159, 182, 185, 192	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → ((pmTrsp‘𝑁)‘{𝑐, 𝑓}):𝑁⟶𝑁)
194	193	ffnd 6661	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) Fn 𝑁)
195		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ ((𝑐 ∈ 𝑁 ∧ 𝑓 ∈ 𝑁) ∧ 𝑐 ≠ 𝑓)) ∧ 𝑎 ∈ 𝑁) → 𝑎 ∈ 𝑁)
196		fnelnfp 7121	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2973	. . . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . . . . . . . . . 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 7371	. . . . . . . . . . . . . . . . . . . . 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 4486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑎 = 𝑓 → if(𝑎 = 𝑓, ((𝑑‘𝑐)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)) = ((𝑑‘𝑎)𝐹𝑏))
204	203	adantl 481	. . . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . . 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 4486	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . . 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 7433	. . . . . . . . . . . . . . 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 580	. . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . 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 6832	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑐 → (𝑑‘𝑎) = (𝑑‘𝑐))
216	215	oveq1d 7371	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑐 → ((𝑑‘𝑎)𝐹𝑏) = ((𝑑‘𝑐)𝐹𝑏))
217		iftrue 4483	. . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑐 → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑐, ((𝑑‘𝑐)𝐹𝑏), if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏))))
219		fveq2 6832	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑓 → (𝑑‘𝑎) = (𝑑‘𝑓))
220	219	oveq1d 7371	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑓 → ((𝑑‘𝑎)𝐹𝑏) = ((𝑑‘𝑓)𝐹𝑏))
221		iftrue 4483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑓 → if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)) = ((𝑑‘𝑓)𝐹𝑏))
222	220, 221	eqtr4d 2772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑓 → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
223	222	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ 𝑎 = 𝑐 ∧ 𝑎 = 𝑓) → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
224		iffalse 4486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝑎 = 𝑓 → if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)) = ((𝑑‘𝑎)𝐹𝑏))
225	224	eqcomd 2740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑎 = 𝑓 → ((𝑑‘𝑎)𝐹𝑏) = if(𝑎 = 𝑓, ((𝑑‘𝑓)𝐹𝑏), ((𝑑‘𝑎)𝐹𝑏)))
226	225	adantl 481	. . . . . . . . . . . . . . . . . . . . 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 4486	. . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . . . . 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 6835	. . . . . . . . . . . . . . 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 6835	. . . . . . . . . . . . . 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 2779	. . . . . . . . . . . 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 6831	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) → (𝑒‘𝑎) = (((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))
238	237	fveq2d 6836	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) → (𝑑‘(𝑒‘𝑎)) = (𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎)))
239	238	oveq1d 7371	. . . . . . . . . . . . . 14 ⊢ (𝑒 = ((pmTrsp‘𝑁)‘{𝑐, 𝑓}) → ((𝑑‘(𝑒‘𝑎))𝐹𝑏) = ((𝑑‘(((pmTrsp‘𝑁)‘{𝑐, 𝑓})‘𝑎))𝐹𝑏))
240	239	mpoeq3dv 7435	. . . . . . . . . . . . 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 6840	. . . . . . . . . . . 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 247	. . . . . . . . . . 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 456	. . . . . . . . . 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 410	. . . . . . . . 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 3191	. . . . . . . 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 1117	. . . . . 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 1164	. . . . 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 2769	. . . 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 480	. . 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 6832	. . . 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 481	. . 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 2734	. . . . . 6 ⊢ (inv_g‘𝑅) = (inv_g‘𝑅)
254	47	3ad2ant1 1133	. . . . . 6 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑅 ∈ Ring)
255	58	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
256	255	3ad2ant1 1133	. . . . . . . 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 20078	. . . . . . . . 9 ⊢ 𝐾 = (Base‘(mulGrp‘𝑅))
258	31, 257	mhmf 18712	. . . . . . . 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 7028	. . . . . 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 1133	. . . . . . 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 7028	. . . . . 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 20237	. . . . 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 20077	. . . . . . . . 9 ⊢ · = (+_g‘(mulGrp‘𝑅))
266	31, 30, 265	mhmlin 18716	. . . . . . . 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 1373	. . . . . . 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 1133	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) ∧ 𝑑 ∈ (Base‘(SymGrp‘𝑁)) ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑁 ∈ Fin)
269		simp3 1138	. . . . . . . . . 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 21550	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑒 ∈ ran (pmTrsp‘𝑁)) → 𝑒 ∈ ((Base‘(SymGrp‘𝑁)) ∖ (pmEven‘𝑁)))
271	268, 269, 270	syl2anc 584	. . . . . . . . 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 2734	. . . . . . . . . 10 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
273		eqid 2734	. . . . . . . . . 10 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁)
274	272, 273, 54, 31, 253	zrhpsgnodpm 21545	. . . . . . . . 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 1373	. . . . . . . 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 7372	. . . . . . 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 20236	. . . . . . 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 2774	. . . . . 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 7371	. . . . 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 2771	. . . 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 480	. . 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 2773	. 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 1137	. . 3 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → 𝐸:𝑁–1-1-onto→𝑁)
284	34, 31	elsymgbas 19301	. . . 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 18751	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 ∀wral 3049 ∃wrex 3058 ∖ cdif 3896 ⊆ wss 3899 ifcif 4477 {csn 4578 {cpr 4580 class class class wbr 5096 I cid 5516 × cxp 5620 dom cdm 5622 ran crn 5623 ↾ cres 5624 ∘ ccom 5626 Fn wfn 6485 ⟶wf 6486 –1-1-onto→wf1o 6489 ‘cfv 6490 (class class class)co 7356 ∈ cmpo 7358 ∘_f cof 7618 2_oc2o 8389 ↑_m cmap 8761 ≈ cen 8878 Fincfn 8881 Basecbs 17134 +_gcplusg 17175 ._rcmulr 17176 0_gc0g 17357 mrClscmrc 17500 Mndcmnd 18657 MndHom cmhm 18704 SubMndcsubmnd 18705 Grpcgrp 18861 inv_gcminusg 18862 SymGrpcsymg 19296 pmTrspcpmtr 19368 pmSgncpsgn 19416 pmEvencevpm 19417 mulGrpcmgp 20073 1_rcur 20114 Ringcrg 20166 ℤRHomczrh 21452 Mat cmat 22349

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-addf 11103 ax-mulf 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1513 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-ot 4587 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8763 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-sup 9343 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-xnn0 12473 df-z 12487 df-dec 12606 df-uz 12750 df-rp 12904 df-fz 13422 df-fzo 13569 df-seq 13923 df-exp 13983 df-hash 14252 df-word 14435 df-lsw 14484 df-concat 14492 df-s1 14518 df-substr 14563 df-pfx 14593 df-splice 14671 df-reverse 14680 df-s2 14769 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-0g 17359 df-gsum 17360 df-prds 17365 df-pws 17367 df-mre 17503 df-mrc 17504 df-acs 17506 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18706 df-submnd 18707 df-efmnd 18792 df-grp 18864 df-minusg 18865 df-mulg 18996 df-subg 19051 df-ghm 19140 df-gim 19186 df-oppg 19273 df-symg 19297 df-pmtr 19369 df-psgn 19418 df-evpm 19419 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-oppr 20271 df-dvdsr 20291 df-unit 20292 df-invr 20322 df-dvr 20335 df-rhm 20406 df-subrng 20477 df-subrg 20501 df-drng 20662 df-sra 21123 df-rgmod 21124 df-cnfld 21308 df-zring 21400 df-zrh 21456 df-dsmm 21685 df-frlm 21700 df-mat 22350

This theorem is referenced by: mdetunilem8 22561

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