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Mirrors > Home > MPE Home > Th. List > ringsubdi | Structured version Visualization version GIF version |
Description: Ring multiplication distributes over subtraction. (subdi 11061 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
Ref | Expression |
---|---|
ringsubdi.b | ⊢ 𝐵 = (Base‘𝑅) |
ringsubdi.t | ⊢ · = (.r‘𝑅) |
ringsubdi.m | ⊢ − = (-g‘𝑅) |
ringsubdi.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
ringsubdi.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ringsubdi.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
ringsubdi.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
ringsubdi | ⊢ (𝜑 → (𝑋 · (𝑌 − 𝑍)) = ((𝑋 · 𝑌) − (𝑋 · 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringsubdi.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | ringsubdi.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | ringsubdi.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | ringgrp 19231 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp) |
6 | ringsubdi.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
7 | ringsubdi.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
8 | eqid 2818 | . . . . . 6 ⊢ (invg‘𝑅) = (invg‘𝑅) | |
9 | 7, 8 | grpinvcl 18089 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((invg‘𝑅)‘𝑍) ∈ 𝐵) |
10 | 5, 6, 9 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((invg‘𝑅)‘𝑍) ∈ 𝐵) |
11 | eqid 2818 | . . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
12 | ringsubdi.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
13 | 7, 11, 12 | ringdi 19245 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ((invg‘𝑅)‘𝑍) ∈ 𝐵)) → (𝑋 · (𝑌(+g‘𝑅)((invg‘𝑅)‘𝑍))) = ((𝑋 · 𝑌)(+g‘𝑅)(𝑋 · ((invg‘𝑅)‘𝑍)))) |
14 | 1, 2, 3, 10, 13 | syl13anc 1364 | . . 3 ⊢ (𝜑 → (𝑋 · (𝑌(+g‘𝑅)((invg‘𝑅)‘𝑍))) = ((𝑋 · 𝑌)(+g‘𝑅)(𝑋 · ((invg‘𝑅)‘𝑍)))) |
15 | 7, 12, 8, 1, 2, 6 | ringmneg2 19276 | . . . 4 ⊢ (𝜑 → (𝑋 · ((invg‘𝑅)‘𝑍)) = ((invg‘𝑅)‘(𝑋 · 𝑍))) |
16 | 15 | oveq2d 7161 | . . 3 ⊢ (𝜑 → ((𝑋 · 𝑌)(+g‘𝑅)(𝑋 · ((invg‘𝑅)‘𝑍))) = ((𝑋 · 𝑌)(+g‘𝑅)((invg‘𝑅)‘(𝑋 · 𝑍)))) |
17 | 14, 16 | eqtrd 2853 | . 2 ⊢ (𝜑 → (𝑋 · (𝑌(+g‘𝑅)((invg‘𝑅)‘𝑍))) = ((𝑋 · 𝑌)(+g‘𝑅)((invg‘𝑅)‘(𝑋 · 𝑍)))) |
18 | ringsubdi.m | . . . . 5 ⊢ − = (-g‘𝑅) | |
19 | 7, 11, 8, 18 | grpsubval 18087 | . . . 4 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) = (𝑌(+g‘𝑅)((invg‘𝑅)‘𝑍))) |
20 | 3, 6, 19 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌(+g‘𝑅)((invg‘𝑅)‘𝑍))) |
21 | 20 | oveq2d 7161 | . 2 ⊢ (𝜑 → (𝑋 · (𝑌 − 𝑍)) = (𝑋 · (𝑌(+g‘𝑅)((invg‘𝑅)‘𝑍)))) |
22 | 7, 12 | ringcl 19240 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
23 | 1, 2, 3, 22 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) |
24 | 7, 12 | ringcl 19240 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 · 𝑍) ∈ 𝐵) |
25 | 1, 2, 6, 24 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑍) ∈ 𝐵) |
26 | 7, 11, 8, 18 | grpsubval 18087 | . . 3 ⊢ (((𝑋 · 𝑌) ∈ 𝐵 ∧ (𝑋 · 𝑍) ∈ 𝐵) → ((𝑋 · 𝑌) − (𝑋 · 𝑍)) = ((𝑋 · 𝑌)(+g‘𝑅)((invg‘𝑅)‘(𝑋 · 𝑍)))) |
27 | 23, 25, 26 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝑋 · 𝑌) − (𝑋 · 𝑍)) = ((𝑋 · 𝑌)(+g‘𝑅)((invg‘𝑅)‘(𝑋 · 𝑍)))) |
28 | 17, 21, 27 | 3eqtr4d 2863 | 1 ⊢ (𝜑 → (𝑋 · (𝑌 − 𝑍)) = ((𝑋 · 𝑌) − (𝑋 · 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 Grpcgrp 18041 invgcminusg 18042 -gcsg 18043 Ringcrg 19226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mgp 19169 df-ur 19181 df-ring 19228 |
This theorem is referenced by: 2idlcpbl 19935 mdetuni0 21158 chfacfpmmulgsum2 21401 nrgdsdi 23201 nrginvrcnlem 23227 ply1divmo 24656 ornglmulle 30805 |
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