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Mirrors > Home > MPE Home > Th. List > opprneg | Structured version Visualization version GIF version |
Description: The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
opprbas.1 | ⊢ 𝑂 = (oppr‘𝑅) |
opprneg.2 | ⊢ 𝑁 = (invg‘𝑅) |
Ref | Expression |
---|---|
opprneg | ⊢ 𝑁 = (invg‘𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2724 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2724 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
3 | eqid 2724 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | opprneg.2 | . . 3 ⊢ 𝑁 = (invg‘𝑅) | |
5 | 1, 2, 3, 4 | grpinvfval 18900 | . 2 ⊢ 𝑁 = (𝑥 ∈ (Base‘𝑅) ↦ (℩𝑦 ∈ (Base‘𝑅)(𝑦(+g‘𝑅)𝑥) = (0g‘𝑅))) |
6 | opprbas.1 | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
7 | 6, 1 | opprbas 20235 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑂) |
8 | 6, 2 | oppradd 20237 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑂) |
9 | 6, 3 | oppr0 20243 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑂) |
10 | eqid 2724 | . . 3 ⊢ (invg‘𝑂) = (invg‘𝑂) | |
11 | 7, 8, 9, 10 | grpinvfval 18900 | . 2 ⊢ (invg‘𝑂) = (𝑥 ∈ (Base‘𝑅) ↦ (℩𝑦 ∈ (Base‘𝑅)(𝑦(+g‘𝑅)𝑥) = (0g‘𝑅))) |
12 | 5, 11 | eqtr4i 2755 | 1 ⊢ 𝑁 = (invg‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ↦ cmpt 5222 ‘cfv 6534 ℩crio 7357 (class class class)co 7402 Basecbs 17145 +gcplusg 17198 0gc0g 17386 invgcminusg 18856 opprcoppr 20227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-plusg 17211 df-mulr 17212 df-0g 17388 df-minusg 18859 df-oppr 20228 |
This theorem is referenced by: unitnegcl 20291 oppreqg 33069 ldualneg 38513 ldualvsub 38519 lcdneg 40975 lcdvsub 40982 |
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