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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 20139 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
14 | 6, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Grp) |
15 | ringgrp 20139 | . . . 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 19146 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆)) |
20 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 19 | isrhm2d 20385 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 +gcplusg 17204 .rcmulr 17205 Grpcgrp 18861 1rcur 20082 Ringcrg 20134 RingHom crh 20367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-0g 17394 df-mhm 18711 df-ghm 19135 df-mgp 20036 df-ur 20083 df-ring 20136 df-rhm 20370 |
This theorem is referenced by: issrngd 20700 evlslem1 21956 frobrhm 32818 imasrhm 32907 evls1maprhm 33214 qqhrhm 33433 rhmmpl 41588 evlsmaprhm 41605 |
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