rngokerinj - Mathbox for Jeff Madsen

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Theorem rngokerinj 37976

Description: A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)

**Hypotheses**
Ref	Expression
rngkerinj.1	⊢ 𝐺 = (1^st ‘𝑅)
rngkerinj.2	⊢ 𝑋 = ran 𝐺
rngkerinj.3	⊢ 𝑊 = (GId‘𝐺)
rngkerinj.4	⊢ 𝐽 = (1^st ‘𝑆)
rngkerinj.5	⊢ 𝑌 = ran 𝐽
rngkerinj.6	⊢ 𝑍 = (GId‘𝐽)

**Assertion**
Ref	Expression
rngokerinj	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹:𝑋–1-1→𝑌 ↔ (^◡𝐹 “ {𝑍}) = {𝑊}))

**Proof of Theorem rngokerinj**
Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2	1	rngogrpo 37911	. . 3 ⊢ (𝑅 ∈ RingOps → (1^st ‘𝑅) ∈ GrpOp)
3	2	3ad2ant1 1134	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (1^st ‘𝑅) ∈ GrpOp)
4		eqid 2737	. . . 4 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
5	4	rngogrpo 37911	. . 3 ⊢ (𝑆 ∈ RingOps → (1^st ‘𝑆) ∈ GrpOp)
6	5	3ad2ant2 1135	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (1^st ‘𝑆) ∈ GrpOp)
7	1, 4	rngogrphom 37972	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹 ∈ ((1^st ‘𝑅) GrpOpHom (1^st ‘𝑆)))
8		rngkerinj.2	. . . 4 ⊢ 𝑋 = ran 𝐺
9		rngkerinj.1	. . . . 5 ⊢ 𝐺 = (1^st ‘𝑅)
10	9	rneqi 5955	. . . 4 ⊢ ran 𝐺 = ran (1^st ‘𝑅)
11	8, 10	eqtri 2765	. . 3 ⊢ 𝑋 = ran (1^st ‘𝑅)
12		rngkerinj.3	. . . 4 ⊢ 𝑊 = (GId‘𝐺)
13	9	fveq2i 6917	. . . 4 ⊢ (GId‘𝐺) = (GId‘(1^st ‘𝑅))
14	12, 13	eqtri 2765	. . 3 ⊢ 𝑊 = (GId‘(1^st ‘𝑅))
15		rngkerinj.5	. . . 4 ⊢ 𝑌 = ran 𝐽
16		rngkerinj.4	. . . . 5 ⊢ 𝐽 = (1^st ‘𝑆)
17	16	rneqi 5955	. . . 4 ⊢ ran 𝐽 = ran (1^st ‘𝑆)
18	15, 17	eqtri 2765	. . 3 ⊢ 𝑌 = ran (1^st ‘𝑆)
19		rngkerinj.6	. . . 4 ⊢ 𝑍 = (GId‘𝐽)
20	16	fveq2i 6917	. . . 4 ⊢ (GId‘𝐽) = (GId‘(1^st ‘𝑆))
21	19, 20	eqtri 2765	. . 3 ⊢ 𝑍 = (GId‘(1^st ‘𝑆))
22	11, 14, 18, 21	grpokerinj 37894	. 2 ⊢ (((1^st ‘𝑅) ∈ GrpOp ∧ (1^st ‘𝑆) ∈ GrpOp ∧ 𝐹 ∈ ((1^st ‘𝑅) GrpOpHom (1^st ‘𝑆))) → (𝐹:𝑋–1-1→𝑌 ↔ (^◡𝐹 “ {𝑍}) = {𝑊}))
23	3, 6, 7, 22	syl3anc 1372	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹:𝑋–1-1→𝑌 ↔ (^◡𝐹 “ {𝑍}) = {𝑊}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1539 ∈ wcel 2108 {csn 4634 ^◡ccnv 5692 ran crn 5694 “ cima 5696 –1-1→wf1 6566 ‘cfv 6569 (class class class)co 7438 1^st c1st 8020 GrpOpcgr 30534 GIdcgi 30535 GrpOpHom cghomOLD 37884 RingOpscrngo 37895 RingOpsHom crngohom 37961

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-op 4641 df-uni 4916 df-iun 5001 df-br 5152 df-opab 5214 df-mpt 5235 df-id 5587 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-riota 7395 df-ov 7441 df-oprab 7442 df-mpo 7443 df-1st 8022 df-2nd 8023 df-map 8876 df-grpo 30538 df-gid 30539 df-ginv 30540 df-gdiv 30541 df-ablo 30590 df-ghomOLD 37885 df-rngo 37896 df-rngohom 37964

This theorem is referenced by: (None)

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