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Mirrors > Home > MPE Home > Th. List > oppgmnd | Structured version Visualization version GIF version |
Description: The opposite of a monoid is a monoid. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) |
Ref | Expression |
---|---|
oppgbas.1 | ⊢ 𝑂 = (oppg‘𝑅) |
Ref | Expression |
---|---|
oppgmnd | ⊢ (𝑅 ∈ Mnd → 𝑂 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppgbas.1 | . . . 4 ⊢ 𝑂 = (oppg‘𝑅) | |
2 | eqid 2731 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | oppgbas 19180 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑂) |
4 | 3 | a1i 11 | . 2 ⊢ (𝑅 ∈ Mnd → (Base‘𝑅) = (Base‘𝑂)) |
5 | eqidd 2732 | . 2 ⊢ (𝑅 ∈ Mnd → (+g‘𝑂) = (+g‘𝑂)) | |
6 | eqid 2731 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
7 | eqid 2731 | . . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
8 | 6, 1, 7 | oppgplus 19177 | . . 3 ⊢ (𝑥(+g‘𝑂)𝑦) = (𝑦(+g‘𝑅)𝑥) |
9 | 2, 6 | mndcl 18610 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(+g‘𝑅)𝑥) ∈ (Base‘𝑅)) |
10 | 9 | 3com23 1126 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(+g‘𝑅)𝑥) ∈ (Base‘𝑅)) |
11 | 8, 10 | eqeltrid 2836 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑂)𝑦) ∈ (Base‘𝑅)) |
12 | simpl 483 | . . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑅 ∈ Mnd) | |
13 | simpr3 1196 | . . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑧 ∈ (Base‘𝑅)) | |
14 | simpr2 1195 | . . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅)) | |
15 | simpr1 1194 | . . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅)) | |
16 | 2, 6 | mndass 18611 | . . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ (𝑧 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅))) → ((𝑧(+g‘𝑅)𝑦)(+g‘𝑅)𝑥) = (𝑧(+g‘𝑅)(𝑦(+g‘𝑅)𝑥))) |
17 | 12, 13, 14, 15, 16 | syl13anc 1372 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑧(+g‘𝑅)𝑦)(+g‘𝑅)𝑥) = (𝑧(+g‘𝑅)(𝑦(+g‘𝑅)𝑥))) |
18 | 17 | eqcomd 2737 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑧(+g‘𝑅)(𝑦(+g‘𝑅)𝑥)) = ((𝑧(+g‘𝑅)𝑦)(+g‘𝑅)𝑥)) |
19 | 8 | oveq1i 7403 | . . . 4 ⊢ ((𝑥(+g‘𝑂)𝑦)(+g‘𝑂)𝑧) = ((𝑦(+g‘𝑅)𝑥)(+g‘𝑂)𝑧) |
20 | 6, 1, 7 | oppgplus 19177 | . . . 4 ⊢ ((𝑦(+g‘𝑅)𝑥)(+g‘𝑂)𝑧) = (𝑧(+g‘𝑅)(𝑦(+g‘𝑅)𝑥)) |
21 | 19, 20 | eqtri 2759 | . . 3 ⊢ ((𝑥(+g‘𝑂)𝑦)(+g‘𝑂)𝑧) = (𝑧(+g‘𝑅)(𝑦(+g‘𝑅)𝑥)) |
22 | 6, 1, 7 | oppgplus 19177 | . . . . 5 ⊢ (𝑦(+g‘𝑂)𝑧) = (𝑧(+g‘𝑅)𝑦) |
23 | 22 | oveq2i 7404 | . . . 4 ⊢ (𝑥(+g‘𝑂)(𝑦(+g‘𝑂)𝑧)) = (𝑥(+g‘𝑂)(𝑧(+g‘𝑅)𝑦)) |
24 | 6, 1, 7 | oppgplus 19177 | . . . 4 ⊢ (𝑥(+g‘𝑂)(𝑧(+g‘𝑅)𝑦)) = ((𝑧(+g‘𝑅)𝑦)(+g‘𝑅)𝑥) |
25 | 23, 24 | eqtri 2759 | . . 3 ⊢ (𝑥(+g‘𝑂)(𝑦(+g‘𝑂)𝑧)) = ((𝑧(+g‘𝑅)𝑦)(+g‘𝑅)𝑥) |
26 | 18, 21, 25 | 3eqtr4g 2796 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥(+g‘𝑂)𝑦)(+g‘𝑂)𝑧) = (𝑥(+g‘𝑂)(𝑦(+g‘𝑂)𝑧))) |
27 | eqid 2731 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
28 | 2, 27 | mndidcl 18617 | . 2 ⊢ (𝑅 ∈ Mnd → (0g‘𝑅) ∈ (Base‘𝑅)) |
29 | 6, 1, 7 | oppgplus 19177 | . . 3 ⊢ ((0g‘𝑅)(+g‘𝑂)𝑥) = (𝑥(+g‘𝑅)(0g‘𝑅)) |
30 | 2, 6, 27 | mndrid 18623 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(0g‘𝑅)) = 𝑥) |
31 | 29, 30 | eqtrid 2783 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑂)𝑥) = 𝑥) |
32 | 6, 1, 7 | oppgplus 19177 | . . 3 ⊢ (𝑥(+g‘𝑂)(0g‘𝑅)) = ((0g‘𝑅)(+g‘𝑅)𝑥) |
33 | 2, 6, 27 | mndlid 18622 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑥) = 𝑥) |
34 | 32, 33 | eqtrid 2783 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑂)(0g‘𝑅)) = 𝑥) |
35 | 4, 5, 11, 26, 28, 31, 34 | ismndd 18624 | 1 ⊢ (𝑅 ∈ Mnd → 𝑂 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 +gcplusg 17179 0gc0g 17367 Mndcmnd 18602 oppgcoppg 19173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-2nd 7958 df-tpos 8193 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-plusg 17192 df-0g 17369 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-oppg 19174 |
This theorem is referenced by: oppgmndb 19186 oppggrp 19188 gsumwrev 19197 gsumzoppg 19771 gsumzinv 19772 oppgtmd 23530 lsmsnorb2 32360 |
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