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Theorem rngokerinj 34309
 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 2825 . . . 4 (1st𝑅) = (1st𝑅)
21rngogrpo 34244 . . 3 (𝑅 ∈ RingOps → (1st𝑅) ∈ GrpOp)
323ad2ant1 1167 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (1st𝑅) ∈ GrpOp)
4 eqid 2825 . . . 4 (1st𝑆) = (1st𝑆)
54rngogrpo 34244 . . 3 (𝑆 ∈ RingOps → (1st𝑆) ∈ GrpOp)
653ad2ant2 1168 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (1st𝑆) ∈ GrpOp)
71, 4rngogrphom 34305 . 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 5584 . . . 4 ran 𝐺 = ran (1st𝑅)
118, 10eqtri 2849 . . 3 𝑋 = ran (1st𝑅)
12 rngkerinj.3 . . . 4 𝑊 = (GId‘𝐺)
139fveq2i 6436 . . . 4 (GId‘𝐺) = (GId‘(1st𝑅))
1412, 13eqtri 2849 . . 3 𝑊 = (GId‘(1st𝑅))
15 rngkerinj.5 . . . 4 𝑌 = ran 𝐽
16 rngkerinj.4 . . . . 5 𝐽 = (1st𝑆)
1716rneqi 5584 . . . 4 ran 𝐽 = ran (1st𝑆)
1815, 17eqtri 2849 . . 3 𝑌 = ran (1st𝑆)
19 rngkerinj.6 . . . 4 𝑍 = (GId‘𝐽)
2016fveq2i 6436 . . . 4 (GId‘𝐽) = (GId‘(1st𝑆))
2119, 20eqtri 2849 . . 3 𝑍 = (GId‘(1st𝑆))
2211, 14, 18, 21grpokerinj 34227 . 2 (((1st𝑅) ∈ GrpOp ∧ (1st𝑆) ∈ GrpOp ∧ 𝐹 ∈ ((1st𝑅) GrpOpHom (1st𝑆))) → (𝐹:𝑋1-1𝑌 ↔ (𝐹 “ {𝑍}) = {𝑊}))
233, 6, 7, 22syl3anc 1494 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 198   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  {csn 4397  ◡ccnv 5341  ran crn 5343   “ cima 5345  –1-1→wf1 6120  ‘cfv 6123  (class class class)co 6905  1st c1st 7426  GrpOpcgr 27888  GIdcgi 27889   GrpOpHom cghomOLD 34217  RingOpscrngo 34228   RngHom crnghom 34294 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-map 8124  df-grpo 27892  df-gid 27893  df-ginv 27894  df-gdiv 27895  df-ablo 27944  df-ghomOLD 34218  df-rngo 34229  df-rngohom 34297 This theorem is referenced by: (None)
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