Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isrnghm Structured version   Visualization version   GIF version

Theorem isrnghm 42732
Description: A function is a non-unital ring homomorphism iff it is a group homomorphism and preserves multiplication. (Contributed by AV, 22-Feb-2020.)
Hypotheses
Ref Expression
isrnghm.b 𝐵 = (Base‘𝑅)
isrnghm.t · = (.r𝑅)
isrnghm.m = (.r𝑆)
Assertion
Ref Expression
isrnghm (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝐹,𝑦
Allowed substitution hints:   · (𝑥,𝑦)   (𝑥,𝑦)

Proof of Theorem isrnghm
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 rnghmrcl 42729 . 2 (𝐹 ∈ (𝑅 RngHomo 𝑆) → (𝑅 ∈ Rng ∧ 𝑆 ∈ Rng))
2 isrnghm.b . . . . 5 𝐵 = (Base‘𝑅)
3 isrnghm.t . . . . 5 · = (.r𝑅)
4 isrnghm.m . . . . 5 = (.r𝑆)
5 eqid 2825 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
6 eqid 2825 . . . . 5 (+g𝑅) = (+g𝑅)
7 eqid 2825 . . . . 5 (+g𝑆) = (+g𝑆)
82, 3, 4, 5, 6, 7rnghmval 42731 . . . 4 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑅 RngHomo 𝑆) = {𝑓 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑅)𝑦)) = ((𝑓𝑥)(+g𝑆)(𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))})
98eleq2d 2892 . . 3 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑅)𝑦)) = ((𝑓𝑥)(+g𝑆)(𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))}))
10 fveq1 6432 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥(+g𝑅)𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
11 fveq1 6432 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
12 fveq1 6432 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
1311, 12oveq12d 6923 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥)(+g𝑆)(𝑓𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
1410, 13eqeq12d 2840 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥(+g𝑅)𝑦)) = ((𝑓𝑥)(+g𝑆)(𝑓𝑦)) ↔ (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))))
15 fveq1 6432 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦)))
1611, 12oveq12d 6923 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥) (𝑓𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
1715, 16eqeq12d 2840 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
1814, 17anbi12d 624 . . . . . 6 (𝑓 = 𝐹 → (((𝑓‘(𝑥(+g𝑅)𝑦)) = ((𝑓𝑥)(+g𝑆)(𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦))) ↔ ((𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
19182ralbidv 3198 . . . . 5 (𝑓 = 𝐹 → (∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑅)𝑦)) = ((𝑓𝑥)(+g𝑆)(𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦))) ↔ ∀𝑥𝐵𝑦𝐵 ((𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
2019elrab 3585 . . . 4 (𝐹 ∈ {𝑓 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑅)𝑦)) = ((𝑓𝑥)(+g𝑆)(𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))} ↔ (𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 ((𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
21 r19.26-2 3275 . . . . . . 7 (∀𝑥𝐵𝑦𝐵 ((𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) ↔ (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
2221anbi2i 616 . . . . . 6 ((𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 ((𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))) ↔ (𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∧ (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
23 anass 462 . . . . . 6 (((𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) ↔ (𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∧ (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
2422, 23bitr4i 270 . . . . 5 ((𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 ((𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))) ↔ ((𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
252, 5, 6, 7isghm 18011 . . . . . . 7 (𝐹 ∈ (𝑅 GrpHom 𝑆) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))))
26 fvex 6446 . . . . . . . . . . 11 (Base‘𝑆) ∈ V
272fvexi 6447 . . . . . . . . . . 11 𝐵 ∈ V
2826, 27pm3.2i 464 . . . . . . . . . 10 ((Base‘𝑆) ∈ V ∧ 𝐵 ∈ V)
29 elmapg 8135 . . . . . . . . . 10 (((Base‘𝑆) ∈ V ∧ 𝐵 ∈ V) → (𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶(Base‘𝑆)))
3028, 29mp1i 13 . . . . . . . . 9 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶(Base‘𝑆)))
3130anbi1d 623 . . . . . . . 8 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))) ↔ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))))
32 rngabl 42717 . . . . . . . . . 10 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
33 ablgrp 18551 . . . . . . . . . 10 (𝑅 ∈ Abel → 𝑅 ∈ Grp)
3432, 33syl 17 . . . . . . . . 9 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
35 rngabl 42717 . . . . . . . . . 10 (𝑆 ∈ Rng → 𝑆 ∈ Abel)
36 ablgrp 18551 . . . . . . . . . 10 (𝑆 ∈ Abel → 𝑆 ∈ Grp)
3735, 36syl 17 . . . . . . . . 9 (𝑆 ∈ Rng → 𝑆 ∈ Grp)
38 ibar 524 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → ((𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))))))
3934, 37, 38syl2an 589 . . . . . . . 8 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))))))
4031, 39bitr2d 272 . . . . . . 7 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))) ↔ (𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))))
4125, 40syl5rbb 276 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))) ↔ 𝐹 ∈ (𝑅 GrpHom 𝑆)))
4241anbi1d 623 . . . . 5 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (((𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
4324, 42syl5bb 275 . . . 4 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 ((𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
4420, 43syl5bb 275 . . 3 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ {𝑓 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑅)𝑦)) = ((𝑓𝑥)(+g𝑆)(𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))} ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
459, 44bitrd 271 . 2 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
461, 45biadanii 857 1 (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386   = wceq 1656  wcel 2164  wral 3117  {crab 3121  Vcvv 3414  wf 6119  cfv 6123  (class class class)co 6905  𝑚 cmap 8122  Basecbs 16222  +gcplusg 16305  .rcmulr 16306  Grpcgrp 17776   GrpHom cghm 18008  Abelcabl 18547  Rngcrng 42714   RngHomo crngh 42725
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-ov 6908  df-oprab 6909  df-mpt2 6910  df-map 8124  df-ghm 18009  df-abl 18549  df-rng0 42715  df-rnghomo 42727
This theorem is referenced by:  isrnghmmul  42733  rnghmghm  42738  rnghmmul  42740  isrnghm2d  42741  zrrnghm  42757
  Copyright terms: Public domain W3C validator