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Mirrors > Home > MPE Home > Th. List > ringmneg2 | Structured version Visualization version GIF version |
Description: Negation of a product in a ring. (mulneg2 11404 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
Ref | Expression |
---|---|
ringneglmul.b | ⊢ 𝐵 = (Base‘𝑅) |
ringneglmul.t | ⊢ · = (.r‘𝑅) |
ringneglmul.n | ⊢ 𝑁 = (invg‘𝑅) |
ringneglmul.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
ringneglmul.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ringneglmul.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
ringmneg2 | ⊢ (𝜑 → (𝑋 · (𝑁‘𝑌)) = (𝑁‘(𝑋 · 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringneglmul.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | ringneglmul.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | ringneglmul.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | ringgrp 19778 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) |
6 | ringneglmul.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
7 | eqid 2740 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
8 | 6, 7 | ringidcl 19797 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
9 | 1, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝐵) |
10 | ringneglmul.n | . . . . 5 ⊢ 𝑁 = (invg‘𝑅) | |
11 | 6, 10 | grpinvcl 18617 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ (1r‘𝑅) ∈ 𝐵) → (𝑁‘(1r‘𝑅)) ∈ 𝐵) |
12 | 5, 9, 11 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑁‘(1r‘𝑅)) ∈ 𝐵) |
13 | ringneglmul.t | . . . 4 ⊢ · = (.r‘𝑅) | |
14 | 6, 13 | ringass 19793 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑁‘(1r‘𝑅)) ∈ 𝐵)) → ((𝑋 · 𝑌) · (𝑁‘(1r‘𝑅))) = (𝑋 · (𝑌 · (𝑁‘(1r‘𝑅))))) |
15 | 1, 2, 3, 12, 14 | syl13anc 1371 | . 2 ⊢ (𝜑 → ((𝑋 · 𝑌) · (𝑁‘(1r‘𝑅))) = (𝑋 · (𝑌 · (𝑁‘(1r‘𝑅))))) |
16 | 6, 13 | ringcl 19790 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
17 | 1, 2, 3, 16 | syl3anc 1370 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) |
18 | 6, 13, 7, 10, 1, 17 | rngnegr 19824 | . 2 ⊢ (𝜑 → ((𝑋 · 𝑌) · (𝑁‘(1r‘𝑅))) = (𝑁‘(𝑋 · 𝑌))) |
19 | 6, 13, 7, 10, 1, 3 | rngnegr 19824 | . . 3 ⊢ (𝜑 → (𝑌 · (𝑁‘(1r‘𝑅))) = (𝑁‘𝑌)) |
20 | 19 | oveq2d 7285 | . 2 ⊢ (𝜑 → (𝑋 · (𝑌 · (𝑁‘(1r‘𝑅)))) = (𝑋 · (𝑁‘𝑌))) |
21 | 15, 18, 20 | 3eqtr3rd 2789 | 1 ⊢ (𝜑 → (𝑋 · (𝑁‘𝑌)) = (𝑁‘(𝑋 · 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ‘cfv 6431 (class class class)co 7269 Basecbs 16902 .rcmulr 16953 Grpcgrp 18567 invgcminusg 18568 1rcur 19727 Ringcrg 19773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10920 ax-resscn 10921 ax-1cn 10922 ax-icn 10923 ax-addcl 10924 ax-addrcl 10925 ax-mulcl 10926 ax-mulrcl 10927 ax-mulcom 10928 ax-addass 10929 ax-mulass 10930 ax-distr 10931 ax-i2m1 10932 ax-1ne0 10933 ax-1rid 10934 ax-rnegex 10935 ax-rrecex 10936 ax-cnre 10937 ax-pre-lttri 10938 ax-pre-lttrn 10939 ax-pre-ltadd 10940 ax-pre-mulgt0 10941 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7702 df-2nd 7819 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-er 8473 df-en 8709 df-dom 8710 df-sdom 8711 df-pnf 11004 df-mnf 11005 df-xr 11006 df-ltxr 11007 df-le 11008 df-sub 11199 df-neg 11200 df-nn 11966 df-2 12028 df-sets 16855 df-slot 16873 df-ndx 16885 df-base 16903 df-plusg 16965 df-0g 17142 df-mgm 18316 df-sgrp 18365 df-mnd 18376 df-grp 18570 df-minusg 18571 df-mgp 19711 df-ur 19728 df-ring 19775 |
This theorem is referenced by: ringm2neg 19827 ringsubdi 19828 cntzsubr 20047 abvneg 20084 lflnegcl 37077 |
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