rngokerinj - Mathbox for Jeff Madsen

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Theorem rngokerinj 35257

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 RngHom 𝑆)) → (𝐹:𝑋–1-1→𝑌 ↔ (^◡𝐹 “ {𝑍}) = {𝑊}))

**Proof of Theorem rngokerinj**
Step	Hyp	Ref	Expression
1		eqid 2824	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2	1	rngogrpo 35192	. . 3 ⊢ (𝑅 ∈ RingOps → (1^st ‘𝑅) ∈ GrpOp)
3	2	3ad2ant1 1129	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (1^st ‘𝑅) ∈ GrpOp)
4		eqid 2824	. . . 4 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
5	4	rngogrpo 35192	. . 3 ⊢ (𝑆 ∈ RingOps → (1^st ‘𝑆) ∈ GrpOp)
6	5	3ad2ant2 1130	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (1^st ‘𝑆) ∈ GrpOp)
7	1, 4	rngogrphom 35253	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 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 5810	. . . 4 ⊢ ran 𝐺 = ran (1^st ‘𝑅)
11	8, 10	eqtri 2847	. . 3 ⊢ 𝑋 = ran (1^st ‘𝑅)
12		rngkerinj.3	. . . 4 ⊢ 𝑊 = (GId‘𝐺)
13	9	fveq2i 6676	. . . 4 ⊢ (GId‘𝐺) = (GId‘(1^st ‘𝑅))
14	12, 13	eqtri 2847	. . 3 ⊢ 𝑊 = (GId‘(1^st ‘𝑅))
15		rngkerinj.5	. . . 4 ⊢ 𝑌 = ran 𝐽
16		rngkerinj.4	. . . . 5 ⊢ 𝐽 = (1^st ‘𝑆)
17	16	rneqi 5810	. . . 4 ⊢ ran 𝐽 = ran (1^st ‘𝑆)
18	15, 17	eqtri 2847	. . 3 ⊢ 𝑌 = ran (1^st ‘𝑆)
19		rngkerinj.6	. . . 4 ⊢ 𝑍 = (GId‘𝐽)
20	16	fveq2i 6676	. . . 4 ⊢ (GId‘𝐽) = (GId‘(1^st ‘𝑆))
21	19, 20	eqtri 2847	. . 3 ⊢ 𝑍 = (GId‘(1^st ‘𝑆))
22	11, 14, 18, 21	grpokerinj 35175	. 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 1367	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:𝑋–1-1→𝑌 ↔ (^◡𝐹 “ {𝑍}) = {𝑊}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 {csn 4570 ^◡ccnv 5557 ran crn 5559 “ cima 5561 –1-1→wf1 6355 ‘cfv 6358 (class class class)co 7159 1^st c1st 7690 GrpOpcgr 28269 GIdcgi 28270 GrpOpHom cghomOLD 35165 RingOpscrngo 35176 RngHom crnghom 35242

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-map 8411 df-grpo 28273 df-gid 28274 df-ginv 28275 df-gdiv 28276 df-ablo 28325 df-ghomOLD 35166 df-rngo 35177 df-rngohom 35245

This theorem is referenced by: (None)

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