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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 19423 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
14 | 6, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Grp) |
15 | ringgrp 19423 | . . . 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 18487 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆)) |
20 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 19 | isrhm2d 19604 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 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 1rcur 19372 Ringcrg 19418 RingHom crh 19588 |
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 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 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-nel 3039 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-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 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-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 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-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-er 8322 df-map 8441 df-en 8558 df-dom 8559 df-sdom 8560 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-nn 11719 df-2 11781 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-plusg 16683 df-0g 16820 df-mhm 18074 df-ghm 18476 df-mgp 19361 df-ur 19373 df-ring 19420 df-rnghom 19591 |
This theorem is referenced by: issrngd 19753 evlslem1 20898 frobrhm 31064 qqhrhm 31511 |
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