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Theorem kerf1hrm 18566
 Description: A ring homomorphism 𝐹 is injective if and only if its kernel is the singleton {𝑁}. (Contributed by Thierry Arnoux, 27-Oct-2017.) (Proof shortened by AV, 24-Oct-2019.)
Hypotheses
Ref Expression
kerf1hrm.a 𝐴 = (Base‘𝑅)
kerf1hrm.b 𝐵 = (Base‘𝑆)
kerf1hrm.n 𝑁 = (0g𝑅)
kerf1hrm.0 0 = (0g𝑆)
Assertion
Ref Expression
kerf1hrm (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐴1-1𝐵 ↔ (𝐹 “ { 0 }) = {𝑁}))

Proof of Theorem kerf1hrm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 472 . . . . . . 7 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥 ∈ (𝐹 “ { 0 })) → (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵))
2 f1fn 6015 . . . . . . . . . . 11 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
32adantl 481 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) → 𝐹 Fn 𝐴)
4 elpreima 6245 . . . . . . . . . 10 (𝐹 Fn 𝐴 → (𝑥 ∈ (𝐹 “ { 0 }) ↔ (𝑥𝐴 ∧ (𝐹𝑥) ∈ { 0 })))
53, 4syl 17 . . . . . . . . 9 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) → (𝑥 ∈ (𝐹 “ { 0 }) ↔ (𝑥𝐴 ∧ (𝐹𝑥) ∈ { 0 })))
65biimpa 500 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥 ∈ (𝐹 “ { 0 })) → (𝑥𝐴 ∧ (𝐹𝑥) ∈ { 0 }))
76simpld 474 . . . . . . 7 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥 ∈ (𝐹 “ { 0 })) → 𝑥𝐴)
86simprd 478 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥 ∈ (𝐹 “ { 0 })) → (𝐹𝑥) ∈ { 0 })
9 fvex 6113 . . . . . . . . 9 (𝐹𝑥) ∈ V
109elsn 4140 . . . . . . . 8 ((𝐹𝑥) ∈ { 0 } ↔ (𝐹𝑥) = 0 )
118, 10sylib 207 . . . . . . 7 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥 ∈ (𝐹 “ { 0 })) → (𝐹𝑥) = 0 )
12 kerf1hrm.a . . . . . . . . . . 11 𝐴 = (Base‘𝑅)
13 kerf1hrm.b . . . . . . . . . . 11 𝐵 = (Base‘𝑆)
14 kerf1hrm.0 . . . . . . . . . . 11 0 = (0g𝑆)
15 kerf1hrm.n . . . . . . . . . . 11 𝑁 = (0g𝑅)
1612, 13, 14, 15f1rhm0to0 18563 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑥𝐴) → ((𝐹𝑥) = 0𝑥 = 𝑁))
1716biimpd 218 . . . . . . . . 9 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑥𝐴) → ((𝐹𝑥) = 0𝑥 = 𝑁))
18173expa 1257 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥𝐴) → ((𝐹𝑥) = 0𝑥 = 𝑁))
1918imp 444 . . . . . . 7 ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥𝐴) ∧ (𝐹𝑥) = 0 ) → 𝑥 = 𝑁)
201, 7, 11, 19syl21anc 1317 . . . . . 6 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥 ∈ (𝐹 “ { 0 })) → 𝑥 = 𝑁)
2120ex 449 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) → (𝑥 ∈ (𝐹 “ { 0 }) → 𝑥 = 𝑁))
22 velsn 4141 . . . . 5 (𝑥 ∈ {𝑁} ↔ 𝑥 = 𝑁)
2321, 22syl6ibr 241 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) → (𝑥 ∈ (𝐹 “ { 0 }) → 𝑥 ∈ {𝑁}))
2423ssrdv 3574 . . 3 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) → (𝐹 “ { 0 }) ⊆ {𝑁})
25 rhmrcl1 18542 . . . . . . 7 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
26 ringgrp 18375 . . . . . . 7 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
2712, 15grpidcl 17273 . . . . . . 7 (𝑅 ∈ Grp → 𝑁𝐴)
2825, 26, 273syl 18 . . . . . 6 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑁𝐴)
29 rhmghm 18548 . . . . . . . 8 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
3015, 14ghmid 17489 . . . . . . . 8 (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹𝑁) = 0 )
3129, 30syl 17 . . . . . . 7 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹𝑁) = 0 )
32 fvex 6113 . . . . . . . 8 (𝐹𝑁) ∈ V
3332elsn 4140 . . . . . . 7 ((𝐹𝑁) ∈ { 0 } ↔ (𝐹𝑁) = 0 )
3431, 33sylibr 223 . . . . . 6 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹𝑁) ∈ { 0 })
3512, 13rhmf 18549 . . . . . . 7 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝐴𝐵)
36 ffn 5958 . . . . . . 7 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
37 elpreima 6245 . . . . . . 7 (𝐹 Fn 𝐴 → (𝑁 ∈ (𝐹 “ { 0 }) ↔ (𝑁𝐴 ∧ (𝐹𝑁) ∈ { 0 })))
3835, 36, 373syl 18 . . . . . 6 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝑁 ∈ (𝐹 “ { 0 }) ↔ (𝑁𝐴 ∧ (𝐹𝑁) ∈ { 0 })))
3928, 34, 38mpbir2and 959 . . . . 5 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑁 ∈ (𝐹 “ { 0 }))
4039snssd 4281 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → {𝑁} ⊆ (𝐹 “ { 0 }))
4140adantr 480 . . 3 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) → {𝑁} ⊆ (𝐹 “ { 0 }))
4224, 41eqssd 3585 . 2 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) → (𝐹 “ { 0 }) = {𝑁})
4335adantr 480 . . 3 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐹 “ { 0 }) = {𝑁}) → 𝐹:𝐴𝐵)
4429adantr 480 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
45 simpr2l 1113 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥𝐴)
46 simpr2r 1114 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑦𝐴)
47 simpr3 1062 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹𝑥) = (𝐹𝑦))
48 eqid 2610 . . . . . . . . . . . 12 (𝐹 “ { 0 }) = (𝐹 “ { 0 })
49 eqid 2610 . . . . . . . . . . . 12 (-g𝑅) = (-g𝑅)
5012, 14, 48, 49ghmeqker 17510 . . . . . . . . . . 11 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝑥(-g𝑅)𝑦) ∈ (𝐹 “ { 0 })))
5150biimpa 500 . . . . . . . . . 10 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥(-g𝑅)𝑦) ∈ (𝐹 “ { 0 }))
5244, 45, 46, 47, 51syl31anc 1321 . . . . . . . . 9 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑥(-g𝑅)𝑦) ∈ (𝐹 “ { 0 }))
53 simpr1 1060 . . . . . . . . 9 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹 “ { 0 }) = {𝑁})
5452, 53eleqtrd 2690 . . . . . . . 8 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑥(-g𝑅)𝑦) ∈ {𝑁})
55 ovex 6577 . . . . . . . . 9 (𝑥(-g𝑅)𝑦) ∈ V
5655elsn 4140 . . . . . . . 8 ((𝑥(-g𝑅)𝑦) ∈ {𝑁} ↔ (𝑥(-g𝑅)𝑦) = 𝑁)
5754, 56sylib 207 . . . . . . 7 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑥(-g𝑅)𝑦) = 𝑁)
5825adantr 480 . . . . . . . . 9 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑅 ∈ Ring)
5958, 26syl 17 . . . . . . . 8 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑅 ∈ Grp)
6012, 15, 49grpsubeq0 17324 . . . . . . . 8 ((𝑅 ∈ Grp ∧ 𝑥𝐴𝑦𝐴) → ((𝑥(-g𝑅)𝑦) = 𝑁𝑥 = 𝑦))
6159, 45, 46, 60syl3anc 1318 . . . . . . 7 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑥(-g𝑅)𝑦) = 𝑁𝑥 = 𝑦))
6257, 61mpbid 221 . . . . . 6 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥 = 𝑦)
63623anassrs 1282 . . . . 5 ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐹 “ { 0 }) = {𝑁}) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 = 𝑦)
6463ex 449 . . . 4 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐹 “ { 0 }) = {𝑁}) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6564ralrimivva 2954 . . 3 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐹 “ { 0 }) = {𝑁}) → ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
66 dff13 6416 . . 3 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6743, 65, 66sylanbrc 695 . 2 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐹 “ { 0 }) = {𝑁}) → 𝐹:𝐴1-1𝐵)
6842, 67impbida 873 1 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐴1-1𝐵 ↔ (𝐹 “ { 0 }) = {𝑁}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  {csn 4125  ◡ccnv 5037   “ cima 5041   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  0gc0g 15923  Grpcgrp 17245  -gcsg 17247   GrpHom cghm 17480  Ringcrg 18370   RingHom crh 18535 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  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  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-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  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-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-minusg 17249  df-sbg 17250  df-ghm 17481  df-mgp 18313  df-ur 18325  df-ring 18372  df-rnghom 18538 This theorem is referenced by:  zrhf1ker  29347
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