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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 2737 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | oppgbas 18743 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑂) |
4 | 3 | a1i 11 | . 2 ⊢ (𝑅 ∈ Mnd → (Base‘𝑅) = (Base‘𝑂)) |
5 | eqidd 2738 | . 2 ⊢ (𝑅 ∈ Mnd → (+g‘𝑂) = (+g‘𝑂)) | |
6 | eqid 2737 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
7 | eqid 2737 | . . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
8 | 6, 1, 7 | oppgplus 18741 | . . 3 ⊢ (𝑥(+g‘𝑂)𝑦) = (𝑦(+g‘𝑅)𝑥) |
9 | 2, 6 | mndcl 18181 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(+g‘𝑅)𝑥) ∈ (Base‘𝑅)) |
10 | 9 | 3com23 1128 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(+g‘𝑅)𝑥) ∈ (Base‘𝑅)) |
11 | 8, 10 | eqeltrid 2842 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑂)𝑦) ∈ (Base‘𝑅)) |
12 | simpl 486 | . . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑅 ∈ Mnd) | |
13 | simpr3 1198 | . . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑧 ∈ (Base‘𝑅)) | |
14 | simpr2 1197 | . . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅)) | |
15 | simpr1 1196 | . . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅)) | |
16 | 2, 6 | mndass 18182 | . . . . 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 1374 | . . . 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 2743 | . . 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 7223 | . . . 4 ⊢ ((𝑥(+g‘𝑂)𝑦)(+g‘𝑂)𝑧) = ((𝑦(+g‘𝑅)𝑥)(+g‘𝑂)𝑧) |
20 | 6, 1, 7 | oppgplus 18741 | . . . 4 ⊢ ((𝑦(+g‘𝑅)𝑥)(+g‘𝑂)𝑧) = (𝑧(+g‘𝑅)(𝑦(+g‘𝑅)𝑥)) |
21 | 19, 20 | eqtri 2765 | . . 3 ⊢ ((𝑥(+g‘𝑂)𝑦)(+g‘𝑂)𝑧) = (𝑧(+g‘𝑅)(𝑦(+g‘𝑅)𝑥)) |
22 | 6, 1, 7 | oppgplus 18741 | . . . . 5 ⊢ (𝑦(+g‘𝑂)𝑧) = (𝑧(+g‘𝑅)𝑦) |
23 | 22 | oveq2i 7224 | . . . 4 ⊢ (𝑥(+g‘𝑂)(𝑦(+g‘𝑂)𝑧)) = (𝑥(+g‘𝑂)(𝑧(+g‘𝑅)𝑦)) |
24 | 6, 1, 7 | oppgplus 18741 | . . . 4 ⊢ (𝑥(+g‘𝑂)(𝑧(+g‘𝑅)𝑦)) = ((𝑧(+g‘𝑅)𝑦)(+g‘𝑅)𝑥) |
25 | 23, 24 | eqtri 2765 | . . 3 ⊢ (𝑥(+g‘𝑂)(𝑦(+g‘𝑂)𝑧)) = ((𝑧(+g‘𝑅)𝑦)(+g‘𝑅)𝑥) |
26 | 18, 21, 25 | 3eqtr4g 2803 | . 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 2737 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
28 | 2, 27 | mndidcl 18188 | . 2 ⊢ (𝑅 ∈ Mnd → (0g‘𝑅) ∈ (Base‘𝑅)) |
29 | 6, 1, 7 | oppgplus 18741 | . . 3 ⊢ ((0g‘𝑅)(+g‘𝑂)𝑥) = (𝑥(+g‘𝑅)(0g‘𝑅)) |
30 | 2, 6, 27 | mndrid 18194 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(0g‘𝑅)) = 𝑥) |
31 | 29, 30 | syl5eq 2790 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑂)𝑥) = 𝑥) |
32 | 6, 1, 7 | oppgplus 18741 | . . 3 ⊢ (𝑥(+g‘𝑂)(0g‘𝑅)) = ((0g‘𝑅)(+g‘𝑅)𝑥) |
33 | 2, 6, 27 | mndlid 18193 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑥) = 𝑥) |
34 | 32, 33 | syl5eq 2790 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑂)(0g‘𝑅)) = 𝑥) |
35 | 4, 5, 11, 26, 28, 31, 34 | ismndd 18195 | 1 ⊢ (𝑅 ∈ Mnd → 𝑂 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 +gcplusg 16802 0gc0g 16944 Mndcmnd 18173 oppgcoppg 18737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-plusg 16815 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-oppg 18738 |
This theorem is referenced by: oppgmndb 18747 oppggrp 18749 gsumwrev 18758 gsumzoppg 19329 gsumzinv 19330 oppgtmd 22994 lsmsnorb2 31294 |
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