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Mirrors > Home > MPE Home > Th. List > isrhmd | Structured version Visualization version GIF version |
Description: Demonstration of ring homomorphism. (Contributed by Stefan O'Rear, 8-Mar-2015.) |
Ref | Expression |
---|---|
isrhmd.b | ⊢ 𝐵 = (Base‘𝑅) |
isrhmd.o | ⊢ 1 = (1r‘𝑅) |
isrhmd.n | ⊢ 𝑁 = (1r‘𝑆) |
isrhmd.t | ⊢ · = (.r‘𝑅) |
isrhmd.u | ⊢ × = (.r‘𝑆) |
isrhmd.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
isrhmd.s | ⊢ (𝜑 → 𝑆 ∈ Ring) |
isrhmd.ho | ⊢ (𝜑 → (𝐹‘ 1 ) = 𝑁) |
isrhmd.ht | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))) |
isrhmd.c | ⊢ 𝐶 = (Base‘𝑆) |
isrhmd.p | ⊢ + = (+g‘𝑅) |
isrhmd.q | ⊢ ⨣ = (+g‘𝑆) |
isrhmd.f | ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) |
isrhmd.hp | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
Ref | Expression |
---|---|
isrhmd | ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isrhmd.b | . 2 ⊢ 𝐵 = (Base‘𝑅) | |
2 | isrhmd.o | . 2 ⊢ 1 = (1r‘𝑅) | |
3 | isrhmd.n | . 2 ⊢ 𝑁 = (1r‘𝑆) | |
4 | isrhmd.t | . 2 ⊢ · = (.r‘𝑅) | |
5 | isrhmd.u | . 2 ⊢ × = (.r‘𝑆) | |
6 | isrhmd.r | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
7 | isrhmd.s | . 2 ⊢ (𝜑 → 𝑆 ∈ Ring) | |
8 | isrhmd.ho | . 2 ⊢ (𝜑 → (𝐹‘ 1 ) = 𝑁) | |
9 | isrhmd.ht | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))) | |
10 | isrhmd.c | . . 3 ⊢ 𝐶 = (Base‘𝑆) | |
11 | isrhmd.p | . . 3 ⊢ + = (+g‘𝑅) | |
12 | isrhmd.q | . . 3 ⊢ ⨣ = (+g‘𝑆) | |
13 | ringgrp 19296 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
14 | 6, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Grp) |
15 | ringgrp 19296 | . . . 4 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ Grp) | |
16 | 7, 15 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ∈ Grp) |
17 | isrhmd.f | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) | |
18 | isrhmd.hp | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) | |
19 | 1, 10, 11, 12, 14, 16, 17, 18 | isghmd 18361 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆)) |
20 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 19 | isrhm2d 19474 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 Grpcgrp 18097 1rcur 19245 Ringcrg 19291 RingHom crh 19458 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-0g 16709 df-mhm 17950 df-ghm 18350 df-mgp 19234 df-ur 19246 df-ring 19293 df-rnghom 19461 |
This theorem is referenced by: issrngd 19626 evlslem1 20289 qqhrhm 31225 |
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