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Mirrors > Home > MPE Home > Th. List > srngmul | Structured version Visualization version GIF version |
Description: The involution function in a star ring distributes over multiplication, with a change in the order of the factors. (Contributed by Mario Carneiro, 6-Oct-2015.) |
Ref | Expression |
---|---|
srngcl.i | ⊢ ∗ = (*𝑟‘𝑅) |
srngcl.b | ⊢ 𝐵 = (Base‘𝑅) |
srngmul.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
srngmul | ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ∗ ‘(𝑋 · 𝑌)) = (( ∗ ‘𝑌) · ( ∗ ‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2772 | . . . . 5 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
2 | eqid 2772 | . . . . 5 ⊢ (*rf‘𝑅) = (*rf‘𝑅) | |
3 | 1, 2 | srngrhm 19334 | . . . 4 ⊢ (𝑅 ∈ *-Ring → (*rf‘𝑅) ∈ (𝑅 RingHom (oppr‘𝑅))) |
4 | srngcl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
5 | srngmul.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
6 | eqid 2772 | . . . . 5 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅)) | |
7 | 4, 5, 6 | rhmmul 19192 | . . . 4 ⊢ (((*rf‘𝑅) ∈ (𝑅 RingHom (oppr‘𝑅)) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((*rf‘𝑅)‘(𝑋 · 𝑌)) = (((*rf‘𝑅)‘𝑋)(.r‘(oppr‘𝑅))((*rf‘𝑅)‘𝑌))) |
8 | 3, 7 | syl3an1 1143 | . . 3 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((*rf‘𝑅)‘(𝑋 · 𝑌)) = (((*rf‘𝑅)‘𝑋)(.r‘(oppr‘𝑅))((*rf‘𝑅)‘𝑌))) |
9 | 4, 5, 1, 6 | opprmul 19089 | . . 3 ⊢ (((*rf‘𝑅)‘𝑋)(.r‘(oppr‘𝑅))((*rf‘𝑅)‘𝑌)) = (((*rf‘𝑅)‘𝑌) · ((*rf‘𝑅)‘𝑋)) |
10 | 8, 9 | syl6eq 2824 | . 2 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((*rf‘𝑅)‘(𝑋 · 𝑌)) = (((*rf‘𝑅)‘𝑌) · ((*rf‘𝑅)‘𝑋))) |
11 | srngring 19335 | . . . 4 ⊢ (𝑅 ∈ *-Ring → 𝑅 ∈ Ring) | |
12 | 4, 5 | ringcl 19024 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
13 | 11, 12 | syl3an1 1143 | . . 3 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
14 | srngcl.i | . . . 4 ⊢ ∗ = (*𝑟‘𝑅) | |
15 | 4, 14, 2 | stafval 19331 | . . 3 ⊢ ((𝑋 · 𝑌) ∈ 𝐵 → ((*rf‘𝑅)‘(𝑋 · 𝑌)) = ( ∗ ‘(𝑋 · 𝑌))) |
16 | 13, 15 | syl 17 | . 2 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((*rf‘𝑅)‘(𝑋 · 𝑌)) = ( ∗ ‘(𝑋 · 𝑌))) |
17 | 4, 14, 2 | stafval 19331 | . . . 4 ⊢ (𝑌 ∈ 𝐵 → ((*rf‘𝑅)‘𝑌) = ( ∗ ‘𝑌)) |
18 | 17 | 3ad2ant3 1115 | . . 3 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((*rf‘𝑅)‘𝑌) = ( ∗ ‘𝑌)) |
19 | 4, 14, 2 | stafval 19331 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → ((*rf‘𝑅)‘𝑋) = ( ∗ ‘𝑋)) |
20 | 19 | 3ad2ant2 1114 | . . 3 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((*rf‘𝑅)‘𝑋) = ( ∗ ‘𝑋)) |
21 | 18, 20 | oveq12d 6988 | . 2 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((*rf‘𝑅)‘𝑌) · ((*rf‘𝑅)‘𝑋)) = (( ∗ ‘𝑌) · ( ∗ ‘𝑋))) |
22 | 10, 16, 21 | 3eqtr3d 2816 | 1 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ∗ ‘(𝑋 · 𝑌)) = (( ∗ ‘𝑌) · ( ∗ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 ‘cfv 6182 (class class class)co 6970 Basecbs 16329 .rcmulr 16412 *𝑟cstv 16413 Ringcrg 19010 opprcoppr 19085 RingHom crh 19177 *rfcstf 19326 *-Ringcsr 19327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-tpos 7688 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-plusg 16424 df-mulr 16425 df-0g 16561 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-mhm 17793 df-ghm 18117 df-mgp 18953 df-ur 18965 df-ring 19012 df-oppr 19086 df-rnghom 19180 df-staf 19328 df-srng 19329 |
This theorem is referenced by: ipassr 20482 |
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