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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 2610 | . . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | 1 | rngogrpo 32879 | . . 3 ⊢ (𝑅 ∈ RingOps → (1st ‘𝑅) ∈ GrpOp) |
3 | 2 | 3ad2ant1 1075 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (1st ‘𝑅) ∈ GrpOp) |
4 | eqid 2610 | . . . 4 ⊢ (1st ‘𝑆) = (1st ‘𝑆) | |
5 | 4 | rngogrpo 32879 | . . 3 ⊢ (𝑆 ∈ RingOps → (1st ‘𝑆) ∈ GrpOp) |
6 | 5 | 3ad2ant2 1076 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (1st ‘𝑆) ∈ GrpOp) |
7 | 1, 4 | rngogrphom 32940 | . 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 5273 | . . . 4 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
11 | 8, 10 | eqtri 2632 | . . 3 ⊢ 𝑋 = ran (1st ‘𝑅) |
12 | rngkerinj.3 | . . . 4 ⊢ 𝑊 = (GId‘𝐺) | |
13 | 9 | fveq2i 6106 | . . . 4 ⊢ (GId‘𝐺) = (GId‘(1st ‘𝑅)) |
14 | 12, 13 | eqtri 2632 | . . 3 ⊢ 𝑊 = (GId‘(1st ‘𝑅)) |
15 | rngkerinj.5 | . . . 4 ⊢ 𝑌 = ran 𝐽 | |
16 | rngkerinj.4 | . . . . 5 ⊢ 𝐽 = (1st ‘𝑆) | |
17 | 16 | rneqi 5273 | . . . 4 ⊢ ran 𝐽 = ran (1st ‘𝑆) |
18 | 15, 17 | eqtri 2632 | . . 3 ⊢ 𝑌 = ran (1st ‘𝑆) |
19 | rngkerinj.6 | . . . 4 ⊢ 𝑍 = (GId‘𝐽) | |
20 | 16 | fveq2i 6106 | . . . 4 ⊢ (GId‘𝐽) = (GId‘(1st ‘𝑆)) |
21 | 19, 20 | eqtri 2632 | . . 3 ⊢ 𝑍 = (GId‘(1st ‘𝑆)) |
22 | 11, 14, 18, 21 | grpokerinj 32862 | . 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 1318 | 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 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {csn 4125 ◡ccnv 5037 ran crn 5039 “ cima 5041 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 GrpOpcgr 26727 GIdcgi 26728 GrpOpHom cghomOLD 32852 RingOpscrngo 32863 RngHom crnghom 32929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-grpo 26731 df-gid 26732 df-ginv 26733 df-gdiv 26734 df-ablo 26783 df-ghomOLD 32853 df-rngo 32864 df-rngohom 32932 |
This theorem is referenced by: (None) |
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