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Theorem rngokerinj 36437
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 = (1st𝑅)
rngkerinj.2 𝑋 = ran 𝐺
rngkerinj.3 𝑊 = (GId‘𝐺)
rngkerinj.4 𝐽 = (1st𝑆)
rngkerinj.5 𝑌 = ran 𝐽
rngkerinj.6 𝑍 = (GId‘𝐽)
Assertion
Ref Expression
rngokerinj ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:𝑋1-1𝑌 ↔ (𝐹 “ {𝑍}) = {𝑊}))

Proof of Theorem rngokerinj
StepHypRef Expression
1 eqid 2737 . . . 4 (1st𝑅) = (1st𝑅)
21rngogrpo 36372 . . 3 (𝑅 ∈ RingOps → (1st𝑅) ∈ GrpOp)
323ad2ant1 1134 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (1st𝑅) ∈ GrpOp)
4 eqid 2737 . . . 4 (1st𝑆) = (1st𝑆)
54rngogrpo 36372 . . 3 (𝑆 ∈ RingOps → (1st𝑆) ∈ GrpOp)
653ad2ant2 1135 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (1st𝑆) ∈ GrpOp)
71, 4rngogrphom 36433 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹 ∈ ((1st𝑅) GrpOpHom (1st𝑆)))
8 rngkerinj.2 . . . 4 𝑋 = ran 𝐺
9 rngkerinj.1 . . . . 5 𝐺 = (1st𝑅)
109rneqi 5893 . . . 4 ran 𝐺 = ran (1st𝑅)
118, 10eqtri 2765 . . 3 𝑋 = ran (1st𝑅)
12 rngkerinj.3 . . . 4 𝑊 = (GId‘𝐺)
139fveq2i 6846 . . . 4 (GId‘𝐺) = (GId‘(1st𝑅))
1412, 13eqtri 2765 . . 3 𝑊 = (GId‘(1st𝑅))
15 rngkerinj.5 . . . 4 𝑌 = ran 𝐽
16 rngkerinj.4 . . . . 5 𝐽 = (1st𝑆)
1716rneqi 5893 . . . 4 ran 𝐽 = ran (1st𝑆)
1815, 17eqtri 2765 . . 3 𝑌 = ran (1st𝑆)
19 rngkerinj.6 . . . 4 𝑍 = (GId‘𝐽)
2016fveq2i 6846 . . . 4 (GId‘𝐽) = (GId‘(1st𝑆))
2119, 20eqtri 2765 . . 3 𝑍 = (GId‘(1st𝑆))
2211, 14, 18, 21grpokerinj 36355 . 2 (((1st𝑅) ∈ GrpOp ∧ (1st𝑆) ∈ GrpOp ∧ 𝐹 ∈ ((1st𝑅) GrpOpHom (1st𝑆))) → (𝐹:𝑋1-1𝑌 ↔ (𝐹 “ {𝑍}) = {𝑊}))
233, 6, 7, 22syl3anc 1372 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:𝑋1-1𝑌 ↔ (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  1-1wf1 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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