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Theorem isrnghm 20458
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 RngHom 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 20455 . 2 (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝑅 ∈ Rng ∧ 𝑆 ∈ Rng))
2 isrnghm.b . . . . 5 𝐵 = (Base‘𝑅)
3 isrnghm.t . . . . 5 · = (.r𝑅)
4 isrnghm.m . . . . 5 = (.r𝑆)
5 eqid 2735 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
6 eqid 2735 . . . . 5 (+g𝑅) = (+g𝑅)
7 eqid 2735 . . . . 5 (+g𝑆) = (+g𝑆)
82, 3, 4, 5, 6, 7rnghmval 20457 . . . 4 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑅 RngHom 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 RngHom 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ ((Base‘𝑆) ↑m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑅)𝑦)) = ((𝑓𝑥)(+g𝑆)(𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))}))
10 fveq1 6906 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥(+g𝑅)𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
11 fveq1 6906 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
12 fveq1 6906 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
1311, 12oveq12d 7449 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥)(+g𝑆)(𝑓𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
1410, 13eqeq12d 2751 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥(+g𝑅)𝑦)) = ((𝑓𝑥)(+g𝑆)(𝑓𝑦)) ↔ (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))))
15 fveq1 6906 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦)))
1611, 12oveq12d 7449 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥) (𝑓𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
1715, 16eqeq12d 2751 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
1814, 17anbi12d 632 . . . . . 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 3219 . . . . 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 3695 . . . 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 3136 . . . . . . 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 623 . . . . . 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 468 . . . . . 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 278 . . . . 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 19246 . . . . . . 7 (𝐹 ∈ (𝑅 GrpHom 𝑆) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))))
26 fvex 6920 . . . . . . . . . . 11 (Base‘𝑆) ∈ V
272fvexi 6921 . . . . . . . . . . 11 𝐵 ∈ V
2826, 27pm3.2i 470 . . . . . . . . . 10 ((Base‘𝑆) ∈ V ∧ 𝐵 ∈ V)
29 elmapg 8878 . . . . . . . . . 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 631 . . . . . . . 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 20173 . . . . . . . . . 10 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
33 ablgrp 19818 . . . . . . . . . 10 (𝑅 ∈ Abel → 𝑅 ∈ Grp)
3432, 33syl 17 . . . . . . . . 9 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
35 rngabl 20173 . . . . . . . . . 10 (𝑆 ∈ Rng → 𝑆 ∈ Abel)
36 ablgrp 19818 . . . . . . . . . 10 (𝑆 ∈ Abel → 𝑆 ∈ Grp)
3735, 36syl 17 . . . . . . . . 9 (𝑆 ∈ Rng → 𝑆 ∈ Grp)
38 ibar 528 . . . . . . . . 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 596 . . . . . . . 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 280 . . . . . . 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, 40bitr2id 284 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))) ↔ 𝐹 ∈ (𝑅 GrpHom 𝑆)))
4241anbi1d 631 . . . . 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, 42bitrid 283 . . . 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, 43bitrid 283 . . 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 279 . 2 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
461, 45biadanii 822 1 (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  Grpcgrp 18964   GrpHom cghm 19243  Abelcabl 19814  Rngcrng 20170   RngHom crnghm 20451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-ghm 19244  df-abl 19816  df-rng 20171  df-rnghm 20453
This theorem is referenced by:  isrnghmmul  20459  rnghmghm  20464  rnghmmul  20466  isrnghm2d  20467  zrrnghm  20553  rngqiprngho  21331
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