fldhmf1 - Mathbox for metakunt

	Mathbox for metakunt		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fldhmf1		Structured version Visualization version GIF version

Theorem fldhmf1 40943

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 20255	. . . 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 20254	. . . . . . . . . . . . . . 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 20318	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
12	10, 11	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
13	12	simpld 495	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ DivRing)
14	13	ad4antr 730	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐾 ∈ DivRing)
15		drnggrp 20317	. . . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . 16 ⊢ (inv_g‘𝐾) = (inv_g‘𝐾)
19	2, 18	grpinvcl 18868	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Grp ∧ 𝑏 ∈ 𝐴) → ((inv_g‘𝐾)‘𝑏) ∈ 𝐴)
20	16, 17, 19	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((inv_g‘𝐾)‘𝑏) ∈ 𝐴)
21		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
22		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝐿) = (+_g‘𝐿)
23	2, 21, 22	ghmlin 19091	. . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . . 17 ⊢ (inv_g‘𝐿) = (inv_g‘𝐿)
26	2, 18, 25	ghminv 19093	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ (𝐾 GrpHom 𝐿) ∧ 𝑏 ∈ 𝐴) → (𝐹‘((inv_g‘𝐾)‘𝑏)) = ((inv_g‘𝐿)‘(𝐹‘𝑏)))
27	8, 17, 26	syl2anc 584	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) → 𝐿 ∈ Field)
33		isfld 20318	. . . . . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) → 𝐿 ∈ DivRing)
36	35	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐿 ∈ DivRing)
37		drngring 20314	. . . . . . . . . . . . . . . . . 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 20058	. . . . . . . . . . . . . . . 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 7084	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘𝑏) ∈ 𝐵)
42		eqid 2732	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝐿) = (0_g‘𝐿)
43	3, 22, 42, 25	grprinv 18871	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ Grp ∧ (𝐹‘𝑏) ∈ 𝐵) → ((𝐹‘𝑏)(+_g‘𝐿)((inv_g‘𝐿)‘(𝐹‘𝑏))) = (0_g‘𝐿))
44	39, 41, 43	syl2anc 584	. . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . 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 2772	. . . . . . . . . . . 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 18823	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Grp ∧ 𝑎 ∈ 𝐴 ∧ ((inv_g‘𝐾)‘𝑏) ∈ 𝐴) → (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ∈ 𝐴)
50	16, 9, 20, 49	syl3anc 1371	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏)) ∈ 𝐴)
51	2, 18	grpinvinv 18886	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ Grp ∧ 𝑏 ∈ 𝐴) → ((inv_g‘𝐾)‘((inv_g‘𝐾)‘𝑏)) = 𝑏)
52	16, 17, 51	syl2anc 584	. . . . . . . . . . . . . . . . . . 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 2996	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝑏 ≠ 𝑎)
55	52, 54	eqnetrd 3008	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ((inv_g‘𝐾)‘((inv_g‘𝐾)‘𝑏)) ≠ 𝑎)
56		eqid 2732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘𝐾) = (0_g‘𝐾)
57	2, 21, 56, 18	grpinvid2 18873	. . . . . . . . . . . . . . . . . . . 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 2985	. . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . . . 18 ⊢ (Unit‘𝐾) = (Unit‘𝐾)
63	2, 62, 56	drngunit 20312	. . . . . . . . . . . . . . . . 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 20282	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 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 20281	. . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘(𝑎(+_g‘𝐾)((inv_g‘𝐾)‘𝑏))) ∈ (Unit‘𝐿))
70		eqid 2732	. . . . . . . . . . . . . . . . . 18 ⊢ (Unit‘𝐿) = (Unit‘𝐿)
71		eqid 2732	. . . . . . . . . . . . . . . . . 18 ⊢ (inv_r‘𝐿) = (inv_r‘𝐿)
72	70, 71	unitinvcl 20196	. . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . 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 20312	. . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . 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 512	. . . . . . . . . . . 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 2732	. . . . . . . . . . . . 13 ⊢ (._r‘𝐿) = (._r‘𝐿)
82	3, 81, 42	ringlz 20100	. . . . . . . . . . . 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 2772	. . . . . . . . . 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 2738	. . . . . . . . 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 496	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ CRing)
87	86	crngringd 20062	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ Ring)
88	87	ad4antr 730	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → 𝐾 ∈ Ring)
89		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (inv_r‘𝐾) = (inv_r‘𝐾)
90	62, 89	unitinvcl 20196	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 2732	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐾) = (Base‘𝐾)
93	92, 62	unitcl 20181	. . . . . . . . . . . . 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 2741	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = 𝐴
95	93, 94	eleqtrdi 2843	. . . . . . . . . . . 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 2732	. . . . . . . . . . . 12 ⊢ (._r‘𝐾) = (._r‘𝐾)
98	2, 97, 81	rhmmul 20256	. . . . . . . . . . 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 1371	. . . . . . . . . 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 2738	. . . . . . . . 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 20314	. . . . . . . . . . . . 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 2732	. . . . . . . . . . . . 13 ⊢ (1_r‘𝐾) = (1_r‘𝐾)
104	62, 89, 97, 103	unitrinv 20200	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 6892	. . . . . . . . . 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 2732	. . . . . . . . . . . 12 ⊢ (1_r‘𝐿) = (1_r‘𝐿)
108	103, 107	rhm1 20259	. . . . . . . . . . 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 2772	. . . . . . . . 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 2776	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (0_g‘𝐿) = (1_r‘𝐿))
112	42, 107	drngunz 20326	. . . . . . . . . . . 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 2996	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) → (0_g‘𝐿) ≠ (1_r‘𝐿))
115	114	adantr 481	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (0_g‘𝐿) ≠ (1_r‘𝐿))
116	115	neneqd 2945	. . . . . . . 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 2947	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) ∧ 𝑎 ≠ 𝑏) → (𝐹‘𝑎) ≠ (𝐹‘𝑏))
119	118	ex 413	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐴) → (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏)))
120	119	ralrimiva 3146	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∀𝑏 ∈ 𝐴 (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏)))
121	120	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏)))
122	5, 121	jca 512	. 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ⟶wf 6536 –1-1→wf1 6537 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 +_gcplusg 17193 ._rcmulr 17194 0_gc0g 17381 Grpcgrp 18815 inv_gcminusg 18816 GrpHom cghm 19083 1_rcur 19998 Ringcrg 20049 CRingccrg 20050 Unitcui 20161 inv_rcinvr 20193 RingHom crh 20240 DivRingcdr 20307 Fieldcfield 20308

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-grp 18818 df-minusg 18819 df-ghm 19084 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-rnghom 20243 df-drng 20309 df-field 20310

This theorem is referenced by: (None)

W3C validator