rngokerinj - Mathbox for Jeff Madsen

	Mathbox for Jeff Madsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > rngokerinj		Structured version Visualization version GIF version

Theorem rngokerinj 37356

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 2726	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2	1	rngogrpo 37291	. . 3 ⊢ (𝑅 ∈ RingOps → (1^st ‘𝑅) ∈ GrpOp)
3	2	3ad2ant1 1130	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (1^st ‘𝑅) ∈ GrpOp)
4		eqid 2726	. . . 4 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
5	4	rngogrpo 37291	. . 3 ⊢ (𝑆 ∈ RingOps → (1^st ‘𝑆) ∈ GrpOp)
6	5	3ad2ant2 1131	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (1^st ‘𝑆) ∈ GrpOp)
7	1, 4	rngogrphom 37352	. 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 5930	. . . 4 ⊢ ran 𝐺 = ran (1^st ‘𝑅)
11	8, 10	eqtri 2754	. . 3 ⊢ 𝑋 = ran (1^st ‘𝑅)
12		rngkerinj.3	. . . 4 ⊢ 𝑊 = (GId‘𝐺)
13	9	fveq2i 6888	. . . 4 ⊢ (GId‘𝐺) = (GId‘(1^st ‘𝑅))
14	12, 13	eqtri 2754	. . 3 ⊢ 𝑊 = (GId‘(1^st ‘𝑅))
15		rngkerinj.5	. . . 4 ⊢ 𝑌 = ran 𝐽
16		rngkerinj.4	. . . . 5 ⊢ 𝐽 = (1^st ‘𝑆)
17	16	rneqi 5930	. . . 4 ⊢ ran 𝐽 = ran (1^st ‘𝑆)
18	15, 17	eqtri 2754	. . 3 ⊢ 𝑌 = ran (1^st ‘𝑆)
19		rngkerinj.6	. . . 4 ⊢ 𝑍 = (GId‘𝐽)
20	16	fveq2i 6888	. . . 4 ⊢ (GId‘𝐽) = (GId‘(1^st ‘𝑆))
21	19, 20	eqtri 2754	. . 3 ⊢ 𝑍 = (GId‘(1^st ‘𝑆))
22	11, 14, 18, 21	grpokerinj 37274	. 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 1368	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 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {csn 4623 ^◡ccnv 5668 ran crn 5670 “ cima 5672 –1-1→wf1 6534 ‘cfv 6537 (class class class)co 7405 1^st c1st 7972 GrpOpcgr 30251 GIdcgi 30252 GrpOpHom cghomOLD 37264 RingOpscrngo 37275 RingOpsHom crngohom 37341

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

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-ral 3056 df-rex 3065 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-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-id 5567 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-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-1st 7974 df-2nd 7975 df-map 8824 df-grpo 30255 df-gid 30256 df-ginv 30257 df-gdiv 30258 df-ablo 30307 df-ghomOLD 37265 df-rngo 37276 df-rngohom 37344

This theorem is referenced by: (None)

W3C validator