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Theorem isrnghm 45450
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 45447 . 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 2738 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
6 eqid 2738 . . . . 5 (+g𝑅) = (+g𝑅)
7 eqid 2738 . . . . 5 (+g𝑆) = (+g𝑆)
82, 3, 4, 5, 6, 7rnghmval 45449 . . . 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 2824 . . 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 6773 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥(+g𝑅)𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
11 fveq1 6773 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
12 fveq1 6773 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
1311, 12oveq12d 7293 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥)(+g𝑆)(𝑓𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
1410, 13eqeq12d 2754 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥(+g𝑅)𝑦)) = ((𝑓𝑥)(+g𝑆)(𝑓𝑦)) ↔ (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))))
15 fveq1 6773 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦)))
1611, 12oveq12d 7293 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥) (𝑓𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
1715, 16eqeq12d 2754 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
1814, 17anbi12d 631 . . . . . 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 3129 . . . . 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 3624 . . . 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 3096 . . . . . . 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 469 . . . . . 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 277 . . . . 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 18834 . . . . . . 7 (𝐹 ∈ (𝑅 GrpHom 𝑆) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))))
26 fvex 6787 . . . . . . . . . . 11 (Base‘𝑆) ∈ V
272fvexi 6788 . . . . . . . . . . 11 𝐵 ∈ V
2826, 27pm3.2i 471 . . . . . . . . . 10 ((Base‘𝑆) ∈ V ∧ 𝐵 ∈ V)
29 elmapg 8628 . . . . . . . . . 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 630 . . . . . . . 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 45435 . . . . . . . . . 10 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
33 ablgrp 19391 . . . . . . . . . 10 (𝑅 ∈ Abel → 𝑅 ∈ Grp)
3432, 33syl 17 . . . . . . . . 9 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
35 rngabl 45435 . . . . . . . . . 10 (𝑆 ∈ Rng → 𝑆 ∈ Abel)
36 ablgrp 19391 . . . . . . . . . 10 (𝑆 ∈ Abel → 𝑆 ∈ Grp)
3735, 36syl 17 . . . . . . . . 9 (𝑆 ∈ Rng → 𝑆 ∈ Grp)
38 ibar 529 . . . . . . . . 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 279 . . . . . . 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 630 . . . . 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 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, 43syl5bb 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 278 . 2 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
461, 45biadanii 819 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 205  wa 396   = wceq 1539  wcel 2106  wral 3064  {crab 3068  Vcvv 3432  wf 6429  cfv 6433  (class class class)co 7275  m cmap 8615  Basecbs 16912  +gcplusg 16962  .rcmulr 16963  Grpcgrp 18577   GrpHom cghm 18831  Abelcabl 19387  Rngcrng 45432   RngHomo crngh 45443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-ghm 18832  df-abl 19389  df-rng0 45433  df-rnghomo 45445
This theorem is referenced by:  isrnghmmul  45451  rnghmghm  45456  rnghmmul  45458  isrnghm2d  45459  zrrnghm  45475
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