rngokerinj - Mathbox for Jeff Madsen

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Theorem rngokerinj 38712

Description: Obsolete theorem, use rhmkerinj 20652 instead. A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

**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 2762	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2	1	rngogrpo 38647	. . 3 ⊢ (𝑅 ∈ RingOps → (1^st ‘𝑅) ∈ GrpOp)
3	2	3ad2ant1 1151	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (1^st ‘𝑅) ∈ GrpOp)
4		eqid 2762	. . . 4 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
5	4	rngogrpo 38647	. . 3 ⊢ (𝑆 ∈ RingOps → (1^st ‘𝑆) ∈ GrpOp)
6	5	3ad2ant2 1152	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (1^st ‘𝑆) ∈ GrpOp)
7	1, 4	rngogrphom 38708	. 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 5925	. . . 4 ⊢ ran 𝐺 = ran (1^st ‘𝑅)
11	8, 10	eqtri 2785	. . 3 ⊢ 𝑋 = ran (1^st ‘𝑅)
12		rngkerinj.3	. . . 4 ⊢ 𝑊 = (GId‘𝐺)
13	9	fveq2i 6885	. . . 4 ⊢ (GId‘𝐺) = (GId‘(1^st ‘𝑅))
14	12, 13	eqtri 2785	. . 3 ⊢ 𝑊 = (GId‘(1^st ‘𝑅))
15		rngkerinj.5	. . . 4 ⊢ 𝑌 = ran 𝐽
16		rngkerinj.4	. . . . 5 ⊢ 𝐽 = (1^st ‘𝑆)
17	16	rneqi 5925	. . . 4 ⊢ ran 𝐽 = ran (1^st ‘𝑆)
18	15, 17	eqtri 2785	. . 3 ⊢ 𝑌 = ran (1^st ‘𝑆)
19		rngkerinj.6	. . . 4 ⊢ 𝑍 = (GId‘𝐽)
20	16	fveq2i 6885	. . . 4 ⊢ (GId‘𝐽) = (GId‘(1^st ‘𝑆))
21	19, 20	eqtri 2785	. . 3 ⊢ 𝑍 = (GId‘(1^st ‘𝑆))
22	11, 14, 18, 21	grpokerinj 38630	. 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 1398	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹:𝑋–1-1→𝑌 ↔ (^◡𝐹 “ {𝑍}) = {𝑊}))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 {csn 4587 ^◡ccnv 5658 ran crn 5660 “ cima 5662 –1-1→wf1 6534 ‘cfv 6537 (class class class)co 7416 1^st c1st 7987 GrpOpcgr 30956 GIdcgi 30957 GrpOpHom cghomOLD 38620 RingOpscrngo 38631 RingOpsHom crngohom 38697

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7989 df-2nd 7990 df-map 8831 df-grpo 30960 df-gid 30961 df-ginv 30962 df-gdiv 30963 df-ablo 31012 df-ghomOLD 38621 df-rngo 38632 df-rngohom 38700

This theorem is used by: (None)

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