fldhmf1 - Mathbox for metakunt

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Theorem fldhmf1 41593

Description: A field homomorphism is injective. This follows immediately from the definition of the ring homomorphism that sends the multiplicative identity to the multiplicative identity. (Contributed by metakunt, 7-Jan-2025.)

**Hypotheses**
Ref	Expression
fldhmf1.1	⊢ (𝜑 → 𝐾 ∈ Field)
fldhmf1.2	⊢ (𝜑 → 𝐿 ∈ Field)
fldhmf1.3	⊢ (𝜑 → 𝐹 ∈ (𝐾 RingHom 𝐿))
fldhmf1.4	⊢ 𝐴 = (Base‘𝐾)
fldhmf1.5	⊢ 𝐵 = (Base‘𝐿)

**Assertion**
Ref	Expression
fldhmf1	⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)

Proof of Theorem fldhmf1 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fldhmf1.3	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐾 RingHom 𝐿))
2		fldhmf1.4	. . . . 5 ⊢ 𝐴 = (Base‘𝐾)
3		fldhmf1.5	. . . . 5 ⊢ 𝐵 = (Base‘𝐿)
4	2, 3	rhmf 20431	. . . 4 ⊢ (𝐹 ∈ (𝐾 RingHom 𝐿) → 𝐹:𝐴⟶𝐵)
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
6	1	ad4antr 730	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐹 ∈ (𝐾 RingHom 𝐿))
7		rhmghm 20430	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐾 RingHom 𝐿) → 𝐹 ∈ (𝐾 GrpHom 𝐿))
8	6, 7	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐹 ∈ (𝐾 GrpHom 𝐿))
9		simp-4r 782	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑎 ∈ 𝐴)
10		fldhmf1.1	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 ∈ Field)
11		isfld 20642	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
12	10, 11	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
13	12	simpld 493	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ DivRing)
14	13	ad4antr 730	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐾 ∈ DivRing)
15		drnggrp 20641	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ DivRing → 𝐾 ∈ Grp)
16	14, 15	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐾 ∈ Grp)
17		simpllr 774	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑏 ∈ 𝐴)
18		eqid 2728	. . . . . . . . . . . . . . . 16 ⊢ (inv_g‘𝐾) = (inv_g‘𝐾)
19	2, 18	grpinvcl 18951	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Grp ∧ 𝑏 ∈ 𝐴) → ((inv_g‘𝐾)‘𝑏) ∈ 𝐴)
20	16, 17, 19	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((inv_g‘𝐾)‘𝑏) ∈ 𝐴)
21		eqid 2728	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
22		eqid 2728	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝐿) = (+_g‘𝐿)
23	2, 21, 22	ghmlin 19182	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (𝐾 GrpHom 𝐿) ∧ 𝑎 ∈ 𝐴 ∧ ((inv_g‘𝐾)‘𝑏) ∈ 𝐴) → (𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))) = ((𝐹‘𝑎)(+_g‘𝐿)(𝐹‘((inv_g‘𝐾)‘𝑏))))
24	8, 9, 20, 23	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))) = ((𝐹‘𝑎)(+_g‘𝐿)(𝐹‘((inv_g‘𝐾)‘𝑏))))
25		eqid 2728	. . . . . . . . . . . . . . . . 17 ⊢ (inv_g‘𝐿) = (inv_g‘𝐿)
26	2, 18, 25	ghminv 19184	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ (𝐾 GrpHom 𝐿) ∧ 𝑏 ∈ 𝐴) → (𝐹‘((inv_g‘𝐾)‘𝑏)) = ((inv_g‘𝐿)‘(𝐹‘𝑏)))
27	8, 17, 26	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘((inv_g‘𝐾)‘𝑏)) = ((inv_g‘𝐿)‘(𝐹‘𝑏)))
28	27	oveq2d 7442	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((𝐹‘𝑎)(+_g‘𝐿)(𝐹‘((inv_g‘𝐾)‘𝑏))) = ((𝐹‘𝑎)(+_g‘𝐿)((inv_g‘𝐿)‘(𝐹‘𝑏))))
29		simpr 483	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘𝑎) = (𝐹‘𝑏))
30	29	oveq1d 7441	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((𝐹‘𝑎)(+_g‘𝐿)((inv_g‘𝐿)‘(𝐹‘𝑏))) = ((𝐹‘𝑏)(+_g‘𝐿)((inv_g‘𝐿)‘(𝐹‘𝑏))))
31		fldhmf1.2	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐿 ∈ Field)
32	31	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) → 𝐿 ∈ Field)
33		isfld 20642	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐿 ∈ Field ↔ (𝐿 ∈ DivRing ∧ 𝐿 ∈ CRing))
34	32, 33	sylib 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) → (𝐿 ∈ DivRing ∧ 𝐿 ∈ CRing))
35	34	simpld 493	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) → 𝐿 ∈ DivRing)
36	35	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐿 ∈ DivRing)
37		drngring 20638	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 ∈ DivRing → 𝐿 ∈ Ring)
38	36, 37	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐿 ∈ Ring)
39	38	ringgrpd 20189	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐿 ∈ Grp)
40	6, 4	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐹:𝐴⟶𝐵)
41	40, 17	ffvelcdmd 7100	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘𝑏) ∈ 𝐵)
42		eqid 2728	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝐿) = (0_g‘𝐿)
43	3, 22, 42, 25	grprinv 18954	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ Grp ∧ (𝐹‘𝑏) ∈ 𝐵) → ((𝐹‘𝑏)(+_g‘𝐿)((inv_g‘𝐿)‘(𝐹‘𝑏))) = (0_g‘𝐿))
44	39, 41, 43	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((𝐹‘𝑏)(+_g‘𝐿)((inv_g‘𝐿)‘(𝐹‘𝑏))) = (0_g‘𝐿))
45	30, 44	eqtrd 2768	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((𝐹‘𝑎)(+_g‘𝐿)((inv_g‘𝐿)‘(𝐹‘𝑏))) = (0_g‘𝐿))
46	28, 45	eqtrd 2768	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((𝐹‘𝑎)(+_g‘𝐿)(𝐹‘((inv_g‘𝐾)‘𝑏))) = (0_g‘𝐿))
47	24, 46	eqtrd 2768	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))) = (0_g‘𝐿))
48	47	oveq1d 7441	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))(._r‘𝐿)(𝐹‘((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))))) = ((0_g‘𝐿)(._r‘𝐿)(𝐹‘((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))))))
49	2, 21	grpcl 18905	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Grp ∧ 𝑎 ∈ 𝐴 ∧ ((inv_g‘𝐾)‘𝑏) ∈ 𝐴) → (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ∈ 𝐴)
50	16, 9, 20, 49	syl3anc 1368	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ∈ 𝐴)
51	2, 18	grpinvinv 18969	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ Grp ∧ 𝑏 ∈ 𝐴) → ((inv_g‘𝐾)‘((inv_g‘𝐾)‘𝑏)) = 𝑏)
52	16, 17, 51	syl2anc 582	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((inv_g‘𝐾)‘((inv_g‘𝐾)‘𝑏)) = 𝑏)
53		simplr 767	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑎 ≠ 𝑏)
54	53	necomd 2993	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑏 ≠ 𝑎)
55	52, 54	eqnetrd 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((inv_g‘𝐾)‘((inv_g‘𝐾)‘𝑏)) ≠ 𝑎)
56		eqid 2728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘𝐾) = (0_g‘𝐾)
57	2, 21, 56, 18	grpinvid2 18956	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ Grp ∧ ((inv_g‘𝐾)‘𝑏) ∈ 𝐴 ∧ 𝑎 ∈ 𝐴) → (((inv_g‘𝐾)‘((inv_g‘𝐾)‘𝑏)) = 𝑎 ↔ (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) = (0_g‘𝐾)))
58	57	necon3bid 2982	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ Grp ∧ ((inv_g‘𝐾)‘𝑏) ∈ 𝐴 ∧ 𝑎 ∈ 𝐴) → (((inv_g‘𝐾)‘((inv_g‘𝐾)‘𝑏)) ≠ 𝑎 ↔ (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ≠ (0_g‘𝐾)))
59	16, 20, 9, 58	syl3anc 1368	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (((inv_g‘𝐾)‘((inv_g‘𝐾)‘𝑏)) ≠ 𝑎 ↔ (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ≠ (0_g‘𝐾)))
60	55, 59	mpbid 231	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ≠ (0_g‘𝐾))
61	50, 60	jca 510	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ∈ 𝐴 ∧ (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ≠ (0_g‘𝐾)))
62		eqid 2728	. . . . . . . . . . . . . . . . . 18 ⊢ (Unit‘𝐾) = (Unit‘𝐾)
63	2, 62, 56	drngunit 20636	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ DivRing → ((𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ∈ (Unit‘𝐾) ↔ ((𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ∈ 𝐴 ∧ (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ≠ (0_g‘𝐾))))
64	14, 63	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ∈ (Unit‘𝐾) ↔ ((𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ∈ 𝐴 ∧ (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ≠ (0_g‘𝐾))))
65	61, 64	mpbird 256	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ∈ (Unit‘𝐾))
66		rhmunitinv 20457	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ (𝐾 RingHom 𝐿) ∧ (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ∈ (Unit‘𝐾)) → (𝐹‘((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) = ((inv_r‘𝐿)‘(𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))))
67	6, 65, 66	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) = ((inv_r‘𝐿)‘(𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))))
68		elrhmunit 20456	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ (𝐾 RingHom 𝐿) ∧ (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ∈ (Unit‘𝐾)) → (𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))) ∈ (Unit‘𝐿))
69	6, 65, 68	syl2anc 582	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))) ∈ (Unit‘𝐿))
70		eqid 2728	. . . . . . . . . . . . . . . . . 18 ⊢ (Unit‘𝐿) = (Unit‘𝐿)
71		eqid 2728	. . . . . . . . . . . . . . . . . 18 ⊢ (inv_r‘𝐿) = (inv_r‘𝐿)
72	70, 71	unitinvcl 20336	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ Ring ∧ (𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))) ∈ (Unit‘𝐿)) → ((inv_r‘𝐿)‘(𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ∈ (Unit‘𝐿))
73	38, 69, 72	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((inv_r‘𝐿)‘(𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ∈ (Unit‘𝐿))
74	3, 70, 42	drngunit 20636	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 ∈ DivRing → (((inv_r‘𝐿)‘(𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ∈ (Unit‘𝐿) ↔ (((inv_r‘𝐿)‘(𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ∈ 𝐵 ∧ ((inv_r‘𝐿)‘(𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ≠ (0_g‘𝐿))))
75	36, 74	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (((inv_r‘𝐿)‘(𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ∈ (Unit‘𝐿) ↔ (((inv_r‘𝐿)‘(𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ∈ 𝐵 ∧ ((inv_r‘𝐿)‘(𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ≠ (0_g‘𝐿))))
76	75	biimpd 228	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (((inv_r‘𝐿)‘(𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ∈ (Unit‘𝐿) → (((inv_r‘𝐿)‘(𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ∈ 𝐵 ∧ ((inv_r‘𝐿)‘(𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ≠ (0_g‘𝐿))))
77	73, 76	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (((inv_r‘𝐿)‘(𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ∈ 𝐵 ∧ ((inv_r‘𝐿)‘(𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ≠ (0_g‘𝐿)))
78	77	simpld 493	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((inv_r‘𝐿)‘(𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ∈ 𝐵)
79	67, 78	eqeltrd 2829	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ∈ 𝐵)
80	38, 79	jca 510	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐿 ∈ Ring ∧ (𝐹‘((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ∈ 𝐵))
81		eqid 2728	. . . . . . . . . . . . 13 ⊢ (._r‘𝐿) = (._r‘𝐿)
82	3, 81, 42	ringlz 20236	. . . . . . . . . . . 12 ⊢ ((𝐿 ∈ Ring ∧ (𝐹‘((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) ∈ 𝐵) → ((0_g‘𝐿)(._r‘𝐿)(𝐹‘((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))))) = (0_g‘𝐿))
83	80, 82	syl 17	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((0_g‘𝐿)(._r‘𝐿)(𝐹‘((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))))) = (0_g‘𝐿))
84	48, 83	eqtrd 2768	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))(._r‘𝐿)(𝐹‘((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))))) = (0_g‘𝐿))
85	84	eqcomd 2734	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (0_g‘𝐿) = ((𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))(._r‘𝐿)(𝐹‘((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))))))
86	12	simprd 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ CRing)
87	86	crngringd 20193	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ Ring)
88	87	ad4antr 730	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐾 ∈ Ring)
89		eqid 2728	. . . . . . . . . . . . . 14 ⊢ (inv_r‘𝐾) = (inv_r‘𝐾)
90	62, 89	unitinvcl 20336	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Ring ∧ (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ∈ (Unit‘𝐾)) → ((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))) ∈ (Unit‘𝐾))
91	88, 65, 90	syl2anc 582	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))) ∈ (Unit‘𝐾))
92		eqid 2728	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐾) = (Base‘𝐾)
93	92, 62	unitcl 20321	. . . . . . . . . . . . 13 ⊢ (((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))) ∈ (Unit‘𝐾) → ((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))) ∈ (Base‘𝐾))
94	2	eqcomi 2737	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = 𝐴
95	93, 94	eleqtrdi 2839	. . . . . . . . . . . 12 ⊢ (((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))) ∈ (Unit‘𝐾) → ((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))) ∈ 𝐴)
96	91, 95	syl 17	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))) ∈ 𝐴)
97		eqid 2728	. . . . . . . . . . . 12 ⊢ (._r‘𝐾) = (._r‘𝐾)
98	2, 97, 81	rhmmul 20432	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐾 RingHom 𝐿) ∧ (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ∈ 𝐴 ∧ ((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))) ∈ 𝐴) → (𝐹‘((𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))(._r‘𝐾)((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))))) = ((𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))(._r‘𝐿)(𝐹‘((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))))))
99	6, 50, 96, 98	syl3anc 1368	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘((𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))(._r‘𝐾)((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))))) = ((𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))(._r‘𝐿)(𝐹‘((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))))))
100	99	eqcomd 2734	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))(._r‘𝐿)(𝐹‘((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))))) = (𝐹‘((𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))(._r‘𝐾)((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))))))
101		drngring 20638	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ DivRing → 𝐾 ∈ Ring)
102	14, 101	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐾 ∈ Ring)
103		eqid 2728	. . . . . . . . . . . . 13 ⊢ (1_r‘𝐾) = (1_r‘𝐾)
104	62, 89, 97, 103	unitrinv 20340	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Ring ∧ (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ∈ (Unit‘𝐾)) → ((𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))(._r‘𝐾)((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) = (1_r‘𝐾))
105	102, 65, 104	syl2anc 582	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))(._r‘𝐾)((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)))) = (1_r‘𝐾))
106	105	fveq2d 6906	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘((𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))(._r‘𝐾)((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))))) = (𝐹‘(1_r‘𝐾)))
107		eqid 2728	. . . . . . . . . . . 12 ⊢ (1_r‘𝐿) = (1_r‘𝐿)
108	103, 107	rhm1 20435	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝐾 RingHom 𝐿) → (𝐹‘(1_r‘𝐾)) = (1_r‘𝐿))
109	6, 108	syl 17	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘(1_r‘𝐾)) = (1_r‘𝐿))
110	106, 109	eqtrd 2768	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘((𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))(._r‘𝐾)((inv_r‘𝐾)‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))))) = (1_r‘𝐿))
111	85, 100, 110	3eqtrd 2772	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (0_g‘𝐿) = (1_r‘𝐿))
112	42, 107	drngunz 20650	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ DivRing → (1_r‘𝐿) ≠ (0_g‘𝐿))
113	35, 112	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) → (1_r‘𝐿) ≠ (0_g‘𝐿))
114	113	necomd 2993	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) → (0_g‘𝐿) ≠ (1_r‘𝐿))
115	114	adantr 479	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (0_g‘𝐿) ≠ (1_r‘𝐿))
116	115	neneqd 2942	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ¬ (0_g‘𝐿) = (1_r‘𝐿))
117	111, 116	pm2.65da 815	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) → ¬ (𝐹‘𝑎) = (𝐹‘𝑏))
118	117	neqned 2944	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) → (𝐹‘𝑎) ≠ (𝐹‘𝑏))
119	118	ex 411	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) → (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏)))
120	119	ralrimiva 3143	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∀𝑏 ∈ 𝐴 (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏)))
121	120	ralrimiva 3143	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏)))
122	5, 121	jca 510	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏))))
123		dff14a 7286	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏))))
124	122, 123	sylibr 233	1 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ⟶wf 6549 –1-1→wf1 6550 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 +_gcplusg 17240 ._rcmulr 17241 0_gc0g 17428 Grpcgrp 18897 inv_gcminusg 18898 GrpHom cghm 19174 1_rcur 20128 Ringcrg 20180 CRingccrg 20181 Unitcui 20301 inv_rcinvr 20333 RingHom crh 20415 DivRingcdr 20631 Fieldcfield 20632

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-grp 18900 df-minusg 18901 df-ghm 19175 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-rhm 20418 df-drng 20633 df-field 20634

This theorem is referenced by: (None)

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