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Mirrors > Home > MPE Home > Th. List > ringm2neg | Structured version Visualization version GIF version |
Description: Double negation of a product in a ring. (mul2neg 10675 analog.) (Contributed by Mario Carneiro, 4-Dec-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 |
---|---|
ringm2neg | ⊢ (𝜑 → ((𝑁‘𝑋) · (𝑁‘𝑌)) = (𝑋 · 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringneglmul.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | ringneglmul.t | . . 3 ⊢ · = (.r‘𝑅) | |
3 | ringneglmul.n | . . 3 ⊢ 𝑁 = (invg‘𝑅) | |
4 | ringneglmul.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
5 | ringneglmul.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | ringgrp 18760 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
7 | 4, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) |
8 | ringneglmul.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | 1, 3 | grpinvcl 17675 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵) |
10 | 7, 8, 9 | syl2anc 573 | . . 3 ⊢ (𝜑 → (𝑁‘𝑌) ∈ 𝐵) |
11 | 1, 2, 3, 4, 5, 10 | ringmneg1 18804 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋) · (𝑁‘𝑌)) = (𝑁‘(𝑋 · (𝑁‘𝑌)))) |
12 | 1, 2, 3, 4, 5, 8 | ringmneg2 18805 | . . 3 ⊢ (𝜑 → (𝑋 · (𝑁‘𝑌)) = (𝑁‘(𝑋 · 𝑌))) |
13 | 12 | fveq2d 6337 | . 2 ⊢ (𝜑 → (𝑁‘(𝑋 · (𝑁‘𝑌))) = (𝑁‘(𝑁‘(𝑋 · 𝑌)))) |
14 | 1, 2 | ringcl 18769 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
15 | 4, 5, 8, 14 | syl3anc 1476 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) |
16 | 1, 3 | grpinvinv 17690 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝑋 · 𝑌) ∈ 𝐵) → (𝑁‘(𝑁‘(𝑋 · 𝑌))) = (𝑋 · 𝑌)) |
17 | 7, 15, 16 | syl2anc 573 | . 2 ⊢ (𝜑 → (𝑁‘(𝑁‘(𝑋 · 𝑌))) = (𝑋 · 𝑌)) |
18 | 11, 13, 17 | 3eqtrd 2809 | 1 ⊢ (𝜑 → ((𝑁‘𝑋) · (𝑁‘𝑌)) = (𝑋 · 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ‘cfv 6030 (class class class)co 6796 Basecbs 16064 .rcmulr 16150 Grpcgrp 17630 invgcminusg 17631 Ringcrg 18755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-er 7900 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-nn 11227 df-2 11285 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-plusg 16162 df-0g 16310 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-grp 17633 df-minusg 17634 df-mgp 18698 df-ur 18710 df-ring 18757 |
This theorem is referenced by: orngsqr 30144 |
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