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Theorem scmatf1 22033

Description: There is a 1-1 function from a ring to any ring of scalar matrices with positive dimension over this ring. (Contributed by AV, 25-Dec-2019.)

**Hypotheses**
Ref	Expression
scmatrhmval.k	⊢ 𝐾 = (Base‘𝑅)
scmatrhmval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
scmatrhmval.o	⊢ 1 = (1_r‘𝐴)
scmatrhmval.t	⊢ ∗ = ( ·_𝑠 ‘𝐴)
scmatrhmval.f	⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 ))
scmatrhmval.c	⊢ 𝐶 = (𝑁 ScMat 𝑅)

**Assertion**
Ref	Expression
scmatf1	⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹:𝐾–1-1→𝐶)

Distinct variable groups: 𝑥,𝐾 𝑥,𝑅 𝑥, 1 𝑥, ∗ 𝑥,𝐶 𝑥,𝑁 Allowed substitution hints: 𝐴(𝑥) 𝐹(𝑥)

Proof of Theorem scmatf1 Dummy variables 𝑦 𝑧 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scmatrhmval.k	. . . 4 ⊢ 𝐾 = (Base‘𝑅)
2		scmatrhmval.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
3		scmatrhmval.o	. . . 4 ⊢ 1 = (1_r‘𝐴)
4		scmatrhmval.t	. . . 4 ⊢ ∗ = ( ·_𝑠 ‘𝐴)
5		scmatrhmval.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 ))
6		scmatrhmval.c	. . . 4 ⊢ 𝐶 = (𝑁 ScMat 𝑅)
7	1, 2, 3, 4, 5, 6	scmatf 22031	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹:𝐾⟶𝐶)
8	7	3adant2 1132	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹:𝐾⟶𝐶)
9		simpr 486	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
10		simpl 484	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) → 𝑦 ∈ 𝐾)
11	1, 2, 3, 4, 5	scmatrhmval 22029	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐾) → (𝐹‘𝑦) = (𝑦 ∗ 1 ))
12	9, 10, 11	syl2an 597	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (𝐹‘𝑦) = (𝑦 ∗ 1 ))
13		simpr 486	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) → 𝑧 ∈ 𝐾)
14	1, 2, 3, 4, 5	scmatrhmval 22029	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑧 ∈ 𝐾) → (𝐹‘𝑧) = (𝑧 ∗ 1 ))
15	9, 13, 14	syl2an 597	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (𝐹‘𝑧) = (𝑧 ∗ 1 ))
16	12, 15	eqeq12d 2749	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ (𝑦 ∗ 1 ) = (𝑧 ∗ 1 )))
17	16	3adantl2 1168	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ (𝑦 ∗ 1 ) = (𝑧 ∗ 1 )))
18	2	matring 21945	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19		eqid 2733	. . . . . . . . . . . 12 ⊢ (Base‘𝐴) = (Base‘𝐴)
20	19, 3	ringidcl 20083	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Ring → 1 ∈ (Base‘𝐴))
21	18, 20	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 ∈ (Base‘𝐴))
22	21, 10	anim12ci 615	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (𝑦 ∈ 𝐾 ∧ 1 ∈ (Base‘𝐴)))
23	1, 2, 19, 4	matvscl 21933	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 1 ∈ (Base‘𝐴))) → (𝑦 ∗ 1 ) ∈ (Base‘𝐴))
24	22, 23	syldan 592	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (𝑦 ∗ 1 ) ∈ (Base‘𝐴))
25	21, 13	anim12ci 615	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (𝑧 ∈ 𝐾 ∧ 1 ∈ (Base‘𝐴)))
26	1, 2, 19, 4	matvscl 21933	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑧 ∈ 𝐾 ∧ 1 ∈ (Base‘𝐴))) → (𝑧 ∗ 1 ) ∈ (Base‘𝐴))
27	25, 26	syldan 592	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (𝑧 ∗ 1 ) ∈ (Base‘𝐴))
28	24, 27	jca 513	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑦 ∗ 1 ) ∈ (Base‘𝐴) ∧ (𝑧 ∗ 1 ) ∈ (Base‘𝐴)))
29	28	3adantl2 1168	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑦 ∗ 1 ) ∈ (Base‘𝐴) ∧ (𝑧 ∗ 1 ) ∈ (Base‘𝐴)))
30	2, 19	eqmat 21926	. . . . . 6 ⊢ (((𝑦 ∗ 1 ) ∈ (Base‘𝐴) ∧ (𝑧 ∗ 1 ) ∈ (Base‘𝐴)) → ((𝑦 ∗ 1 ) = (𝑧 ∗ 1 ) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗)))
31	29, 30	syl 17	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑦 ∗ 1 ) = (𝑧 ∗ 1 ) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗)))
32		difsnid 4814	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝑁 → ((𝑁 ∖ {𝑖}) ∪ {𝑖}) = 𝑁)
33	32	eqcomd 2739	. . . . . . . . . . 11 ⊢ (𝑖 ∈ 𝑁 → 𝑁 = ((𝑁 ∖ {𝑖}) ∪ {𝑖}))
34	33	adantl 483	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) ∧ 𝑖 ∈ 𝑁) → 𝑁 = ((𝑁 ∖ {𝑖}) ∪ {𝑖}))
35	34	raleqdv 3326	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) ∧ 𝑖 ∈ 𝑁) → (∀𝑗 ∈ 𝑁 (𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ↔ ∀𝑗 ∈ ((𝑁 ∖ {𝑖}) ∪ {𝑖})(𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗)))
36		oveq2 7417	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → (𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑦 ∗ 1 )𝑖))
37		oveq2 7417	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → (𝑖(𝑧 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑖))
38	36, 37	eqeq12d 2749	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → ((𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ↔ (𝑖(𝑦 ∗ 1 )𝑖) = (𝑖(𝑧 ∗ 1 )𝑖)))
39	38	ralunsn 4895	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝑁 → (∀𝑗 ∈ ((𝑁 ∖ {𝑖}) ∪ {𝑖})(𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ↔ (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ∧ (𝑖(𝑦 ∗ 1 )𝑖) = (𝑖(𝑧 ∗ 1 )𝑖))))
40	39	adantl 483	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) ∧ 𝑖 ∈ 𝑁) → (∀𝑗 ∈ ((𝑁 ∖ {𝑖}) ∪ {𝑖})(𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ↔ (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ∧ (𝑖(𝑦 ∗ 1 )𝑖) = (𝑖(𝑧 ∗ 1 )𝑖))))
41	10	anim2i 618	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ 𝐾))
42		df-3an 1090	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ 𝐾))
43	41, 42	sylibr 233	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐾))
44		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ 𝑁 → 𝑖 ∈ 𝑁)
45	44, 44	jca 513	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝑁 → (𝑖 ∈ 𝑁 ∧ 𝑖 ∈ 𝑁))
46		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
47	2, 1, 46, 3, 4	scmatscmide 22009	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑖 ∈ 𝑁)) → (𝑖(𝑦 ∗ 1 )𝑖) = if(𝑖 = 𝑖, 𝑦, (0_g‘𝑅)))
48	43, 45, 47	syl2an 597	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) ∧ 𝑖 ∈ 𝑁) → (𝑖(𝑦 ∗ 1 )𝑖) = if(𝑖 = 𝑖, 𝑦, (0_g‘𝑅)))
49		eqid 2733	. . . . . . . . . . . . 13 ⊢ 𝑖 = 𝑖
50	49	iftruei 4536	. . . . . . . . . . . 12 ⊢ if(𝑖 = 𝑖, 𝑦, (0_g‘𝑅)) = 𝑦
51	48, 50	eqtrdi 2789	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) ∧ 𝑖 ∈ 𝑁) → (𝑖(𝑦 ∗ 1 )𝑖) = 𝑦)
52	13	anim2i 618	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑧 ∈ 𝐾))
53		df-3an 1090	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑧 ∈ 𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑧 ∈ 𝐾))
54	52, 53	sylibr 233	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑧 ∈ 𝐾))
55	2, 1, 46, 3, 4	scmatscmide 22009	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑧 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑖 ∈ 𝑁)) → (𝑖(𝑧 ∗ 1 )𝑖) = if(𝑖 = 𝑖, 𝑧, (0_g‘𝑅)))
56	54, 45, 55	syl2an 597	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) ∧ 𝑖 ∈ 𝑁) → (𝑖(𝑧 ∗ 1 )𝑖) = if(𝑖 = 𝑖, 𝑧, (0_g‘𝑅)))
57	49	iftruei 4536	. . . . . . . . . . . 12 ⊢ if(𝑖 = 𝑖, 𝑧, (0_g‘𝑅)) = 𝑧
58	56, 57	eqtrdi 2789	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) ∧ 𝑖 ∈ 𝑁) → (𝑖(𝑧 ∗ 1 )𝑖) = 𝑧)
59	51, 58	eqeq12d 2749	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) ∧ 𝑖 ∈ 𝑁) → ((𝑖(𝑦 ∗ 1 )𝑖) = (𝑖(𝑧 ∗ 1 )𝑖) ↔ 𝑦 = 𝑧))
60	59	anbi2d 630	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) ∧ 𝑖 ∈ 𝑁) → ((∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ∧ (𝑖(𝑦 ∗ 1 )𝑖) = (𝑖(𝑧 ∗ 1 )𝑖)) ↔ (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ∧ 𝑦 = 𝑧)))
61	35, 40, 60	3bitrd 305	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) ∧ 𝑖 ∈ 𝑁) → (∀𝑗 ∈ 𝑁 (𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ↔ (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ∧ 𝑦 = 𝑧)))
62	61	ralbidva 3176	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ↔ ∀𝑖 ∈ 𝑁 (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ∧ 𝑦 = 𝑧)))
63	62	3adantl2 1168	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ↔ ∀𝑖 ∈ 𝑁 (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ∧ 𝑦 = 𝑧)))
64		r19.26 3112	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝑁 (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ∧ 𝑦 = 𝑧) ↔ (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ∧ ∀𝑖 ∈ 𝑁 𝑦 = 𝑧))
65		rspn0 4353	. . . . . . . . . . 11 ⊢ (𝑁 ≠ ∅ → (∀𝑖 ∈ 𝑁 𝑦 = 𝑧 → 𝑦 = 𝑧))
66	65	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → (∀𝑖 ∈ 𝑁 𝑦 = 𝑧 → 𝑦 = 𝑧))
67	66	adantr 482	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (∀𝑖 ∈ 𝑁 𝑦 = 𝑧 → 𝑦 = 𝑧))
68	67	com12 32	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝑁 𝑦 = 𝑧 → (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → 𝑦 = 𝑧))
69	64, 68	simplbiim 506	. . . . . . 7 ⊢ (∀𝑖 ∈ 𝑁 (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ∧ 𝑦 = 𝑧) → (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → 𝑦 = 𝑧))
70	69	com12 32	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (∀𝑖 ∈ 𝑁 (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧))
71	63, 70	sylbid 239	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖(𝑦 ∗ 1 )𝑗) = (𝑖(𝑧 ∗ 1 )𝑗) → 𝑦 = 𝑧))
72	31, 71	sylbid 239	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑦 ∗ 1 ) = (𝑧 ∗ 1 ) → 𝑦 = 𝑧))
73	17, 72	sylbid 239	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
74	73	ralrimivva 3201	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐾 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
75		dff13 7254	. 2 ⊢ (𝐹:𝐾–1-1→𝐶 ↔ (𝐹:𝐾⟶𝐶 ∧ ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐾 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)))
76	8, 74, 75	sylanbrc 584	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹:𝐾–1-1→𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∖ cdif 3946 ∪ cun 3947 ∅c0 4323 ifcif 4529 {csn 4629 ↦ cmpt 5232 ⟶wf 6540 –1-1→wf1 6541 ‘cfv 6544 (class class class)co 7409 Fincfn 8939 Basecbs 17144 ·_𝑠 cvsca 17201 0_gc0g 17385 1_rcur 20004 Ringcrg 20056 Mat cmat 21907 ScMat cscmat 21991

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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-fzo 13628 df-seq 13967 df-hash 14291 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-hom 17221 df-cco 17222 df-0g 17387 df-gsum 17388 df-prds 17393 df-pws 17395 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-submnd 18672 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mulg 18951 df-subg 19003 df-ghm 19090 df-cntz 19181 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-subrg 20317 df-lmod 20473 df-lss 20543 df-sra 20785 df-rgmod 20786 df-dsmm 21287 df-frlm 21302 df-mamu 21886 df-mat 21908 df-scmat 21993

This theorem is referenced by: scmatf1o 22034

W3C validator