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Mirrors > Home > MPE Home > Th. List > srngadd | Structured version Visualization version GIF version |
Description: The involution function in a star ring distributes over addition. (Contributed by Mario Carneiro, 6-Oct-2015.) |
Ref | Expression |
---|---|
srngcl.i | ⊢ ∗ = (*𝑟‘𝑅) |
srngcl.b | ⊢ 𝐵 = (Base‘𝑅) |
srngadd.p | ⊢ + = (+g‘𝑅) |
Ref | Expression |
---|---|
srngadd | ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ∗ ‘(𝑋 + 𝑌)) = (( ∗ ‘𝑋) + ( ∗ ‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . . 5 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
2 | eqid 2821 | . . . . 5 ⊢ (*rf‘𝑅) = (*rf‘𝑅) | |
3 | 1, 2 | srngrhm 19616 | . . . 4 ⊢ (𝑅 ∈ *-Ring → (*rf‘𝑅) ∈ (𝑅 RingHom (oppr‘𝑅))) |
4 | rhmghm 19471 | . . . 4 ⊢ ((*rf‘𝑅) ∈ (𝑅 RingHom (oppr‘𝑅)) → (*rf‘𝑅) ∈ (𝑅 GrpHom (oppr‘𝑅))) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ *-Ring → (*rf‘𝑅) ∈ (𝑅 GrpHom (oppr‘𝑅))) |
6 | srngcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
7 | srngadd.p | . . . 4 ⊢ + = (+g‘𝑅) | |
8 | 1, 7 | oppradd 19374 | . . . 4 ⊢ + = (+g‘(oppr‘𝑅)) |
9 | 6, 7, 8 | ghmlin 18357 | . . 3 ⊢ (((*rf‘𝑅) ∈ (𝑅 GrpHom (oppr‘𝑅)) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((*rf‘𝑅)‘(𝑋 + 𝑌)) = (((*rf‘𝑅)‘𝑋) + ((*rf‘𝑅)‘𝑌))) |
10 | 5, 9 | syl3an1 1159 | . 2 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((*rf‘𝑅)‘(𝑋 + 𝑌)) = (((*rf‘𝑅)‘𝑋) + ((*rf‘𝑅)‘𝑌))) |
11 | srngring 19617 | . . . 4 ⊢ (𝑅 ∈ *-Ring → 𝑅 ∈ Ring) | |
12 | 6, 7 | ringacl 19322 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
13 | 11, 12 | syl3an1 1159 | . . 3 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
14 | srngcl.i | . . . 4 ⊢ ∗ = (*𝑟‘𝑅) | |
15 | 6, 14, 2 | stafval 19613 | . . 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 | 6, 14, 2 | stafval 19613 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → ((*rf‘𝑅)‘𝑋) = ( ∗ ‘𝑋)) |
18 | 17 | 3ad2ant2 1130 | . . 3 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((*rf‘𝑅)‘𝑋) = ( ∗ ‘𝑋)) |
19 | 6, 14, 2 | stafval 19613 | . . . 4 ⊢ (𝑌 ∈ 𝐵 → ((*rf‘𝑅)‘𝑌) = ( ∗ ‘𝑌)) |
20 | 19 | 3ad2ant3 1131 | . . 3 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((*rf‘𝑅)‘𝑌) = ( ∗ ‘𝑌)) |
21 | 18, 20 | oveq12d 7168 | . 2 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((*rf‘𝑅)‘𝑋) + ((*rf‘𝑅)‘𝑌)) = (( ∗ ‘𝑋) + ( ∗ ‘𝑌))) |
22 | 10, 16, 21 | 3eqtr3d 2864 | 1 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ∗ ‘(𝑋 + 𝑌)) = (( ∗ ‘𝑋) + ( ∗ ‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 *𝑟cstv 16561 GrpHom cghm 18349 Ringcrg 19291 opprcoppr 19366 RingHom crh 19458 *rfcstf 19608 *-Ringcsr 19609 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-tpos 7886 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-3 11695 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-mulr 16573 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-grp 18100 df-ghm 18350 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-rnghom 19461 df-staf 19610 df-srng 19611 |
This theorem is referenced by: ipdi 20778 |
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