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Theorem isrnghm 42420
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 42417 . 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 2760 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
6 eqid 2760 . . . . 5 (+g𝑅) = (+g𝑅)
7 eqid 2760 . . . . 5 (+g𝑆) = (+g𝑆)
82, 3, 4, 5, 6, 7rnghmval 42419 . . . 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 2825 . . 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 6352 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥(+g𝑅)𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
11 fveq1 6352 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
12 fveq1 6352 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
1311, 12oveq12d 6832 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥)(+g𝑆)(𝑓𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
1410, 13eqeq12d 2775 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥(+g𝑅)𝑦)) = ((𝑓𝑥)(+g𝑆)(𝑓𝑦)) ↔ (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))))
15 fveq1 6352 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦)))
1611, 12oveq12d 6832 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥) (𝑓𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
1715, 16eqeq12d 2775 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
1814, 17anbi12d 749 . . . . . 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 3127 . . . . 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 3504 . . . 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 3203 . . . . . . 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 732 . . . . . 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 684 . . . . . 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 267 . . . . 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 17881 . . . . . . 7 (𝐹 ∈ (𝑅 GrpHom 𝑆) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))))
26 fvex 6363 . . . . . . . . . . 11 (Base‘𝑆) ∈ V
27 fvex 6363 . . . . . . . . . . . 12 (Base‘𝑅) ∈ V
282, 27eqeltri 2835 . . . . . . . . . . 11 𝐵 ∈ V
2926, 28pm3.2i 470 . . . . . . . . . 10 ((Base‘𝑆) ∈ V ∧ 𝐵 ∈ V)
30 elmapg 8038 . . . . . . . . . 10 (((Base‘𝑆) ∈ V ∧ 𝐵 ∈ V) → (𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶(Base‘𝑆)))
3129, 30mp1i 13 . . . . . . . . 9 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶(Base‘𝑆)))
3231anbi1d 743 . . . . . . . 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)))))
33 rngabl 42405 . . . . . . . . . 10 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
34 ablgrp 18418 . . . . . . . . . 10 (𝑅 ∈ Abel → 𝑅 ∈ Grp)
3533, 34syl 17 . . . . . . . . 9 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
36 rngabl 42405 . . . . . . . . . 10 (𝑆 ∈ Rng → 𝑆 ∈ Abel)
37 ablgrp 18418 . . . . . . . . . 10 (𝑆 ∈ Abel → 𝑆 ∈ Grp)
3836, 37syl 17 . . . . . . . . 9 (𝑆 ∈ Rng → 𝑆 ∈ Grp)
39 ibar 526 . . . . . . . . 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))))))
4035, 38, 39syl2an 495 . . . . . . . 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))))))
4132, 40bitr2d 269 . . . . . . 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)))))
4225, 41syl5rbb 273 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹 ∈ ((Base‘𝑆) ↑𝑚 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))) ↔ 𝐹 ∈ (𝑅 GrpHom 𝑆)))
4342anbi1d 743 . . . . 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)))))
4424, 43syl5bb 272 . . . 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)))))
4520, 44syl5bb 272 . . 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)))))
469, 45bitrd 268 . 2 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
471, 46biadan2 677 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 196  wa 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  wf 6045  cfv 6049  (class class class)co 6814  𝑚 cmap 8025  Basecbs 16079  +gcplusg 16163  .rcmulr 16164  Grpcgrp 17643   GrpHom cghm 17878  Abelcabl 18414  Rngcrng 42402   RngHomo crngh 42413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027  df-ghm 17879  df-abl 18416  df-rng0 42403  df-rnghomo 42415
This theorem is referenced by:  isrnghmmul  42421  rnghmghm  42426  rnghmmul  42428  isrnghm2d  42429  zrrnghm  42445
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