fldhmf1 - Mathbox for metakunt

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Theorem fldhmf1 41471

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 20387	. . . 4 ⊢ (𝐹 ∈ (𝐾 RingHom 𝐿) → 𝐹:𝐴⟶𝐵)
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
6	1	ad4antr 729	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐹 ∈ (𝐾 RingHom 𝐿))
7		rhmghm 20386	. . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑎 ∈ 𝐴)
10		fldhmf1.1	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 ∈ Field)
11		isfld 20598	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
12	10, 11	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
13	12	simpld 494	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ DivRing)
14	13	ad4antr 729	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐾 ∈ DivRing)
15		drnggrp 20597	. . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑏 ∈ 𝐴)
18		eqid 2726	. . . . . . . . . . . . . . . 16 ⊢ (inv_g‘𝐾) = (inv_g‘𝐾)
19	2, 18	grpinvcl 18917	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Grp ∧ 𝑏 ∈ 𝐴) → ((inv_g‘𝐾)‘𝑏) ∈ 𝐴)
20	16, 17, 19	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((inv_g‘𝐾)‘𝑏) ∈ 𝐴)
21		eqid 2726	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
22		eqid 2726	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝐿) = (+_g‘𝐿)
23	2, 21, 22	ghmlin 19146	. . . . . . . . . . . . . 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 2726	. . . . . . . . . . . . . . . . 17 ⊢ (inv_g‘𝐿) = (inv_g‘𝐿)
26	2, 18, 25	ghminv 19148	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ (𝐾 GrpHom 𝐿) ∧ 𝑏 ∈ 𝐴) → (𝐹‘((inv_g‘𝐾)‘𝑏)) = ((inv_g‘𝐿)‘(𝐹‘𝑏)))
27	8, 17, 26	syl2anc 583	. . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘𝑎) = (𝐹‘𝑏))
30	29	oveq1d 7420	. . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) → 𝐿 ∈ Field)
33		isfld 20598	. . . . . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) → 𝐿 ∈ DivRing)
36	35	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐿 ∈ DivRing)
37		drngring 20594	. . . . . . . . . . . . . . . . . 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 20147	. . . . . . . . . . . . . . . 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 7081	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘𝑏) ∈ 𝐵)
42		eqid 2726	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝐿) = (0_g‘𝐿)
43	3, 22, 42, 25	grprinv 18920	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ Grp ∧ (𝐹‘𝑏) ∈ 𝐵) → ((𝐹‘𝑏)(+_g‘𝐿)((inv_g‘𝐿)‘(𝐹‘𝑏))) = (0_g‘𝐿))
44	39, 41, 43	syl2anc 583	. . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 7420	. . . . . . . . . . 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 18871	. . . . . . . . . . . . . . . . . 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 18935	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ Grp ∧ 𝑏 ∈ 𝐴) → ((inv_g‘𝐾)‘((inv_g‘𝐾)‘𝑏)) = 𝑏)
52	16, 17, 51	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((inv_g‘𝐾)‘((inv_g‘𝐾)‘𝑏)) = 𝑏)
53		simplr 766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑎 ≠ 𝑏)
54	53	necomd 2990	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑏 ≠ 𝑎)
55	52, 54	eqnetrd 3002	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((inv_g‘𝐾)‘((inv_g‘𝐾)‘𝑏)) ≠ 𝑎)
56		eqid 2726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘𝐾) = (0_g‘𝐾)
57	2, 21, 56, 18	grpinvid2 18922	. . . . . . . . . . . . . . . . . . . 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 2979	. . . . . . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . . . 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 2726	. . . . . . . . . . . . . . . . . 18 ⊢ (Unit‘𝐾) = (Unit‘𝐾)
63	2, 62, 56	drngunit 20592	. . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ∈ (Unit‘𝐾))
66		rhmunitinv 20413	. . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . 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 20412	. . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))) ∈ (Unit‘𝐿))
70		eqid 2726	. . . . . . . . . . . . . . . . . 18 ⊢ (Unit‘𝐿) = (Unit‘𝐿)
71		eqid 2726	. . . . . . . . . . . . . . . . . 18 ⊢ (inv_r‘𝐿) = (inv_r‘𝐿)
72	70, 71	unitinvcl 20292	. . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . 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 20592	. . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . 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 511	. . . . . . . . . . . 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 2726	. . . . . . . . . . . . 13 ⊢ (._r‘𝐿) = (._r‘𝐿)
82	3, 81, 42	ringlz 20192	. . . . . . . . . . . 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 2766	. . . . . . . . . 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 2732	. . . . . . . . 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 495	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ CRing)
87	86	crngringd 20151	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ Ring)
88	87	ad4antr 729	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐾 ∈ Ring)
89		eqid 2726	. . . . . . . . . . . . . 14 ⊢ (inv_r‘𝐾) = (inv_r‘𝐾)
90	62, 89	unitinvcl 20292	. . . . . . . . . . . . 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 583	. . . . . . . . . . . 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 2726	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐾) = (Base‘𝐾)
93	92, 62	unitcl 20277	. . . . . . . . . . . . 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 2735	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = 𝐴
95	93, 94	eleqtrdi 2837	. . . . . . . . . . . 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 2726	. . . . . . . . . . . 12 ⊢ (._r‘𝐾) = (._r‘𝐾)
98	2, 97, 81	rhmmul 20388	. . . . . . . . . . 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 2732	. . . . . . . . 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 20594	. . . . . . . . . . . . 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 2726	. . . . . . . . . . . . 13 ⊢ (1_r‘𝐾) = (1_r‘𝐾)
104	62, 89, 97, 103	unitrinv 20296	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 6889	. . . . . . . . . 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 2726	. . . . . . . . . . . 12 ⊢ (1_r‘𝐿) = (1_r‘𝐿)
108	103, 107	rhm1 20391	. . . . . . . . . . 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 2766	. . . . . . . . 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 2770	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (0_g‘𝐿) = (1_r‘𝐿))
112	42, 107	drngunz 20606	. . . . . . . . . . . 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 2990	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) → (0_g‘𝐿) ≠ (1_r‘𝐿))
115	114	adantr 480	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (0_g‘𝐿) ≠ (1_r‘𝐿))
116	115	neneqd 2939	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ¬ (0_g‘𝐿) = (1_r‘𝐿))
117	111, 116	pm2.65da 814	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) → ¬ (𝐹‘𝑎) = (𝐹‘𝑏))
118	117	neqned 2941	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) → (𝐹‘𝑎) ≠ (𝐹‘𝑏))
119	118	ex 412	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) → (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏)))
120	119	ralrimiva 3140	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∀𝑏 ∈ 𝐴 (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏)))
121	120	ralrimiva 3140	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏)))
122	5, 121	jca 511	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏))))
123		dff14a 7265	. 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 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 ⟶wf 6533 –1-1→wf1 6534 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 +_gcplusg 17206 ._rcmulr 17207 0_gc0g 17394 Grpcgrp 18863 inv_gcminusg 18864 GrpHom cghm 19138 1_rcur 20086 Ringcrg 20138 CRingccrg 20139 Unitcui 20257 inv_rcinvr 20289 RingHom crh 20371 DivRingcdr 20587 Fieldcfield 20588

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-grp 18866 df-minusg 18867 df-ghm 19139 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-rhm 20374 df-drng 20589 df-field 20590

This theorem is referenced by: (None)

W3C validator