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Mirrors > Home > MPE Home > Th. List > ringsubdi | Structured version Visualization version GIF version |
Description: Ring multiplication distributes over subtraction. (subdi 10463 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 18552 | . . . . . 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 2622 | . . . . . 6 ⊢ (invg‘𝑅) = (invg‘𝑅) | |
9 | 7, 8 | grpinvcl 17467 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((invg‘𝑅)‘𝑍) ∈ 𝐵) |
10 | 5, 6, 9 | syl2anc 693 | . . . 4 ⊢ (𝜑 → ((invg‘𝑅)‘𝑍) ∈ 𝐵) |
11 | eqid 2622 | . . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
12 | ringsubdi.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
13 | 7, 11, 12 | ringdi 18566 | . . . 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 1328 | . . 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 18597 | . . . 4 ⊢ (𝜑 → (𝑋 · ((invg‘𝑅)‘𝑍)) = ((invg‘𝑅)‘(𝑋 · 𝑍))) |
16 | 15 | oveq2d 6666 | . . 3 ⊢ (𝜑 → ((𝑋 · 𝑌)(+g‘𝑅)(𝑋 · ((invg‘𝑅)‘𝑍))) = ((𝑋 · 𝑌)(+g‘𝑅)((invg‘𝑅)‘(𝑋 · 𝑍)))) |
17 | 14, 16 | eqtrd 2656 | . 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 17465 | . . . 4 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) = (𝑌(+g‘𝑅)((invg‘𝑅)‘𝑍))) |
20 | 3, 6, 19 | syl2anc 693 | . . 3 ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌(+g‘𝑅)((invg‘𝑅)‘𝑍))) |
21 | 20 | oveq2d 6666 | . 2 ⊢ (𝜑 → (𝑋 · (𝑌 − 𝑍)) = (𝑋 · (𝑌(+g‘𝑅)((invg‘𝑅)‘𝑍)))) |
22 | 7, 12 | ringcl 18561 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
23 | 1, 2, 3, 22 | syl3anc 1326 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) |
24 | 7, 12 | ringcl 18561 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 · 𝑍) ∈ 𝐵) |
25 | 1, 2, 6, 24 | syl3anc 1326 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑍) ∈ 𝐵) |
26 | 7, 11, 8, 18 | grpsubval 17465 | . . 3 ⊢ (((𝑋 · 𝑌) ∈ 𝐵 ∧ (𝑋 · 𝑍) ∈ 𝐵) → ((𝑋 · 𝑌) − (𝑋 · 𝑍)) = ((𝑋 · 𝑌)(+g‘𝑅)((invg‘𝑅)‘(𝑋 · 𝑍)))) |
27 | 23, 25, 26 | syl2anc 693 | . 2 ⊢ (𝜑 → ((𝑋 · 𝑌) − (𝑋 · 𝑍)) = ((𝑋 · 𝑌)(+g‘𝑅)((invg‘𝑅)‘(𝑋 · 𝑍)))) |
28 | 17, 21, 27 | 3eqtr4d 2666 | 1 ⊢ (𝜑 → (𝑋 · (𝑌 − 𝑍)) = ((𝑋 · 𝑌) − (𝑋 · 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 .rcmulr 15942 Grpcgrp 17422 invgcminusg 17423 -gcsg 17424 Ringcrg 18547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mgp 18490 df-ur 18502 df-ring 18549 |
This theorem is referenced by: 2idlcpbl 19234 mdetuni0 20427 chfacfpmmulgsum2 20670 nrgdsdi 22469 nrginvrcnlem 22495 ply1divmo 23895 ornglmulle 29805 |
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