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Theorem isrnghmd 45023
Description: Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.)
Hypotheses
Ref Expression
isrnghmd.b 𝐵 = (Base‘𝑅)
isrnghmd.t · = (.r𝑅)
isrnghmd.u × = (.r𝑆)
isrnghmd.r (𝜑𝑅 ∈ Rng)
isrnghmd.s (𝜑𝑆 ∈ Rng)
isrnghmd.ht ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))
isrnghmd.c 𝐶 = (Base‘𝑆)
isrnghmd.p + = (+g𝑅)
isrnghmd.q = (+g𝑆)
isrnghmd.f (𝜑𝐹:𝐵𝐶)
isrnghmd.hp ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
Assertion
Ref Expression
isrnghmd (𝜑𝐹 ∈ (𝑅 RngHomo 𝑆))
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝑥, + ,𝑦   𝑥, ,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   · (𝑥,𝑦)   × (𝑥,𝑦)

Proof of Theorem isrnghmd
StepHypRef Expression
1 isrnghmd.b . 2 𝐵 = (Base‘𝑅)
2 isrnghmd.t . 2 · = (.r𝑅)
3 isrnghmd.u . 2 × = (.r𝑆)
4 isrnghmd.r . 2 (𝜑𝑅 ∈ Rng)
5 isrnghmd.s . 2 (𝜑𝑆 ∈ Rng)
6 isrnghmd.ht . 2 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))
7 isrnghmd.c . . 3 𝐶 = (Base‘𝑆)
8 isrnghmd.p . . 3 + = (+g𝑅)
9 isrnghmd.q . . 3 = (+g𝑆)
10 rngabl 44998 . . . 4 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
11 ablgrp 19031 . . . 4 (𝑅 ∈ Abel → 𝑅 ∈ Grp)
124, 10, 113syl 18 . . 3 (𝜑𝑅 ∈ Grp)
13 rngabl 44998 . . . 4 (𝑆 ∈ Rng → 𝑆 ∈ Abel)
14 ablgrp 19031 . . . 4 (𝑆 ∈ Abel → 𝑆 ∈ Grp)
155, 13, 143syl 18 . . 3 (𝜑𝑆 ∈ Grp)
16 isrnghmd.f . . 3 (𝜑𝐹:𝐵𝐶)
17 isrnghmd.hp . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
181, 7, 8, 9, 12, 15, 16, 17isghmd 18487 . 2 (𝜑𝐹 ∈ (𝑅 GrpHom 𝑆))
191, 2, 3, 4, 5, 6, 18isrnghm2d 45022 1 (𝜑𝐹 ∈ (𝑅 RngHomo 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  wf 6335  cfv 6339  (class class class)co 7172  Basecbs 16588  +gcplusg 16670  .rcmulr 16671  Grpcgrp 18221  Abelcabl 19027  Rngcrng 44995   RngHomo crngh 45006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7175  df-oprab 7176  df-mpo 7177  df-map 8441  df-ghm 18476  df-abl 19029  df-rng0 44996  df-rnghomo 45008
This theorem is referenced by: (None)
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