rngokerinj - Mathbox for Jeff Madsen

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Theorem rngokerinj 38307

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 38242	. . 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 38242	. . 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 38303	. 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 5884	. . . 4 ⊢ ran 𝐺 = ran (1^st ‘𝑅)
11	8, 10	eqtri 2760	. . 3 ⊢ 𝑋 = ran (1^st ‘𝑅)
12		rngkerinj.3	. . . 4 ⊢ 𝑊 = (GId‘𝐺)
13	9	fveq2i 6835	. . . 4 ⊢ (GId‘𝐺) = (GId‘(1^st ‘𝑅))
14	12, 13	eqtri 2760	. . 3 ⊢ 𝑊 = (GId‘(1^st ‘𝑅))
15		rngkerinj.5	. . . 4 ⊢ 𝑌 = ran 𝐽
16		rngkerinj.4	. . . . 5 ⊢ 𝐽 = (1^st ‘𝑆)
17	16	rneqi 5884	. . . 4 ⊢ ran 𝐽 = ran (1^st ‘𝑆)
18	15, 17	eqtri 2760	. . 3 ⊢ 𝑌 = ran (1^st ‘𝑆)
19		rngkerinj.6	. . . 4 ⊢ 𝑍 = (GId‘𝐽)
20	16	fveq2i 6835	. . . 4 ⊢ (GId‘𝐽) = (GId‘(1^st ‘𝑆))
21	19, 20	eqtri 2760	. . 3 ⊢ 𝑍 = (GId‘(1^st ‘𝑆))
22	11, 14, 18, 21	grpokerinj 38225	. 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 1374	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 1542 ∈ wcel 2114 {csn 4568 ^◡ccnv 5621 ran crn 5623 “ cima 5625 –1-1→wf1 6487 ‘cfv 6490 (class class class)co 7358 1^st c1st 7931 GrpOpcgr 30580 GIdcgi 30581 GrpOpHom cghomOLD 38215 RingOpscrngo 38226 RingOpsHom crngohom 38292

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-map 8766 df-grpo 30584 df-gid 30585 df-ginv 30586 df-gdiv 30587 df-ablo 30636 df-ghomOLD 38216 df-rngo 38227 df-rngohom 38295

This theorem is referenced by: (None)

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