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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngokerinj | Structured version Visualization version GIF version |
Description: A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.) |
Ref | Expression |
---|---|
rngkerinj.1 | ⊢ 𝐺 = (1st ‘𝑅) |
rngkerinj.2 | ⊢ 𝑋 = ran 𝐺 |
rngkerinj.3 | ⊢ 𝑊 = (GId‘𝐺) |
rngkerinj.4 | ⊢ 𝐽 = (1st ‘𝑆) |
rngkerinj.5 | ⊢ 𝑌 = ran 𝐽 |
rngkerinj.6 | ⊢ 𝑍 = (GId‘𝐽) |
Ref | Expression |
---|---|
rngokerinj | ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:𝑋–1-1→𝑌 ↔ (◡𝐹 “ {𝑍}) = {𝑊})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | 1 | rngogrpo 35182 | . . 3 ⊢ (𝑅 ∈ RingOps → (1st ‘𝑅) ∈ GrpOp) |
3 | 2 | 3ad2ant1 1129 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (1st ‘𝑅) ∈ GrpOp) |
4 | eqid 2821 | . . . 4 ⊢ (1st ‘𝑆) = (1st ‘𝑆) | |
5 | 4 | rngogrpo 35182 | . . 3 ⊢ (𝑆 ∈ RingOps → (1st ‘𝑆) ∈ GrpOp) |
6 | 5 | 3ad2ant2 1130 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (1st ‘𝑆) ∈ GrpOp) |
7 | 1, 4 | rngogrphom 35243 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹 ∈ ((1st ‘𝑅) GrpOpHom (1st ‘𝑆))) |
8 | rngkerinj.2 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
9 | rngkerinj.1 | . . . . 5 ⊢ 𝐺 = (1st ‘𝑅) | |
10 | 9 | rneqi 5802 | . . . 4 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
11 | 8, 10 | eqtri 2844 | . . 3 ⊢ 𝑋 = ran (1st ‘𝑅) |
12 | rngkerinj.3 | . . . 4 ⊢ 𝑊 = (GId‘𝐺) | |
13 | 9 | fveq2i 6668 | . . . 4 ⊢ (GId‘𝐺) = (GId‘(1st ‘𝑅)) |
14 | 12, 13 | eqtri 2844 | . . 3 ⊢ 𝑊 = (GId‘(1st ‘𝑅)) |
15 | rngkerinj.5 | . . . 4 ⊢ 𝑌 = ran 𝐽 | |
16 | rngkerinj.4 | . . . . 5 ⊢ 𝐽 = (1st ‘𝑆) | |
17 | 16 | rneqi 5802 | . . . 4 ⊢ ran 𝐽 = ran (1st ‘𝑆) |
18 | 15, 17 | eqtri 2844 | . . 3 ⊢ 𝑌 = ran (1st ‘𝑆) |
19 | rngkerinj.6 | . . . 4 ⊢ 𝑍 = (GId‘𝐽) | |
20 | 16 | fveq2i 6668 | . . . 4 ⊢ (GId‘𝐽) = (GId‘(1st ‘𝑆)) |
21 | 19, 20 | eqtri 2844 | . . 3 ⊢ 𝑍 = (GId‘(1st ‘𝑆)) |
22 | 11, 14, 18, 21 | grpokerinj 35165 | . 2 ⊢ (((1st ‘𝑅) ∈ GrpOp ∧ (1st ‘𝑆) ∈ GrpOp ∧ 𝐹 ∈ ((1st ‘𝑅) GrpOpHom (1st ‘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 1533 ∈ wcel 2110 {csn 4561 ◡ccnv 5549 ran crn 5551 “ cima 5553 –1-1→wf1 6347 ‘cfv 6350 (class class class)co 7150 1st c1st 7681 GrpOpcgr 28260 GIdcgi 28261 GrpOpHom cghomOLD 35155 RingOpscrngo 35166 RngHom crnghom 35232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-map 8402 df-grpo 28264 df-gid 28265 df-ginv 28266 df-gdiv 28267 df-ablo 28316 df-ghomOLD 35156 df-rngo 35167 df-rngohom 35235 |
This theorem is referenced by: (None) |
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