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Mathbox for Jeff Madsen |
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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 2737 | . . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | 1 | rngogrpo 36372 | . . 3 ⊢ (𝑅 ∈ RingOps → (1st ‘𝑅) ∈ GrpOp) |
3 | 2 | 3ad2ant1 1134 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (1st ‘𝑅) ∈ GrpOp) |
4 | eqid 2737 | . . . 4 ⊢ (1st ‘𝑆) = (1st ‘𝑆) | |
5 | 4 | rngogrpo 36372 | . . 3 ⊢ (𝑆 ∈ RingOps → (1st ‘𝑆) ∈ GrpOp) |
6 | 5 | 3ad2ant2 1135 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (1st ‘𝑆) ∈ GrpOp) |
7 | 1, 4 | rngogrphom 36433 | . 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 5893 | . . . 4 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
11 | 8, 10 | eqtri 2765 | . . 3 ⊢ 𝑋 = ran (1st ‘𝑅) |
12 | rngkerinj.3 | . . . 4 ⊢ 𝑊 = (GId‘𝐺) | |
13 | 9 | fveq2i 6846 | . . . 4 ⊢ (GId‘𝐺) = (GId‘(1st ‘𝑅)) |
14 | 12, 13 | eqtri 2765 | . . 3 ⊢ 𝑊 = (GId‘(1st ‘𝑅)) |
15 | rngkerinj.5 | . . . 4 ⊢ 𝑌 = ran 𝐽 | |
16 | rngkerinj.4 | . . . . 5 ⊢ 𝐽 = (1st ‘𝑆) | |
17 | 16 | rneqi 5893 | . . . 4 ⊢ ran 𝐽 = ran (1st ‘𝑆) |
18 | 15, 17 | eqtri 2765 | . . 3 ⊢ 𝑌 = ran (1st ‘𝑆) |
19 | rngkerinj.6 | . . . 4 ⊢ 𝑍 = (GId‘𝐽) | |
20 | 16 | fveq2i 6846 | . . . 4 ⊢ (GId‘𝐽) = (GId‘(1st ‘𝑆)) |
21 | 19, 20 | eqtri 2765 | . . 3 ⊢ 𝑍 = (GId‘(1st ‘𝑆)) |
22 | 11, 14, 18, 21 | grpokerinj 36355 | . 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 1372 | 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 205 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {csn 4587 ◡ccnv 5633 ran crn 5635 “ cima 5637 –1-1→wf1 6494 ‘cfv 6497 (class class class)co 7358 1st c1st 7920 GrpOpcgr 29434 GIdcgi 29435 GrpOpHom cghomOLD 36345 RingOpscrngo 36356 RngHom crnghom 36422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-map 8768 df-grpo 29438 df-gid 29439 df-ginv 29440 df-gdiv 29441 df-ablo 29490 df-ghomOLD 36346 df-rngo 36357 df-rngohom 36425 |
This theorem is referenced by: (None) |
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