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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 11697 analog.) Compared with rngmneg2 20185, the proof is shorter making use of the existence of a ring unity. (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 20255 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) |
6 | ringneglmul.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
7 | eqid 2734 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
8 | 6, 7 | ringidcl 20279 | . . . . 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 19017 | . . . 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 20270 | . . 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 20267 | . . . 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 | ringnegr 20316 | . 2 ⊢ (𝜑 → ((𝑋 · 𝑌) · (𝑁‘(1r‘𝑅))) = (𝑁‘(𝑋 · 𝑌))) |
19 | 6, 13, 7, 10, 1, 3 | ringnegr 20316 | . . 3 ⊢ (𝜑 → (𝑌 · (𝑁‘(1r‘𝑅))) = (𝑁‘𝑌)) |
20 | 19 | oveq2d 7446 | . 2 ⊢ (𝜑 → (𝑋 · (𝑌 · (𝑁‘(1r‘𝑅)))) = (𝑋 · (𝑁‘𝑌))) |
21 | 15, 18, 20 | 3eqtr3rd 2783 | 1 ⊢ (𝜑 → (𝑋 · (𝑁‘𝑌)) = (𝑁‘(𝑋 · 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 .rcmulr 17298 Grpcgrp 18963 invgcminusg 18964 1rcur 20198 Ringcrg 20250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 |
This theorem is referenced by: cntzsubr 20622 abvneg 20843 erler 33251 zrhcntr 33941 lflnegcl 39056 |
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