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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 2610 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | oppgbas 17604 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑂) |
4 | 3 | a1i 11 | . 2 ⊢ (𝑅 ∈ Mnd → (Base‘𝑅) = (Base‘𝑂)) |
5 | eqidd 2611 | . 2 ⊢ (𝑅 ∈ Mnd → (+g‘𝑂) = (+g‘𝑂)) | |
6 | eqid 2610 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
7 | eqid 2610 | . . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
8 | 6, 1, 7 | oppgplus 17602 | . . 3 ⊢ (𝑥(+g‘𝑂)𝑦) = (𝑦(+g‘𝑅)𝑥) |
9 | 2, 6 | mndcl 17124 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(+g‘𝑅)𝑥) ∈ (Base‘𝑅)) |
10 | 9 | 3com23 1263 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(+g‘𝑅)𝑥) ∈ (Base‘𝑅)) |
11 | 8, 10 | syl5eqel 2692 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑂)𝑦) ∈ (Base‘𝑅)) |
12 | simpl 472 | . . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑅 ∈ Mnd) | |
13 | simpr3 1062 | . . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑧 ∈ (Base‘𝑅)) | |
14 | simpr2 1061 | . . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅)) | |
15 | simpr1 1060 | . . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅)) | |
16 | 2, 6 | mndass 17125 | . . . . 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 1320 | . . . 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 2616 | . . 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 6559 | . . . 4 ⊢ ((𝑥(+g‘𝑂)𝑦)(+g‘𝑂)𝑧) = ((𝑦(+g‘𝑅)𝑥)(+g‘𝑂)𝑧) |
20 | 6, 1, 7 | oppgplus 17602 | . . . 4 ⊢ ((𝑦(+g‘𝑅)𝑥)(+g‘𝑂)𝑧) = (𝑧(+g‘𝑅)(𝑦(+g‘𝑅)𝑥)) |
21 | 19, 20 | eqtri 2632 | . . 3 ⊢ ((𝑥(+g‘𝑂)𝑦)(+g‘𝑂)𝑧) = (𝑧(+g‘𝑅)(𝑦(+g‘𝑅)𝑥)) |
22 | 6, 1, 7 | oppgplus 17602 | . . . . 5 ⊢ (𝑦(+g‘𝑂)𝑧) = (𝑧(+g‘𝑅)𝑦) |
23 | 22 | oveq2i 6560 | . . . 4 ⊢ (𝑥(+g‘𝑂)(𝑦(+g‘𝑂)𝑧)) = (𝑥(+g‘𝑂)(𝑧(+g‘𝑅)𝑦)) |
24 | 6, 1, 7 | oppgplus 17602 | . . . 4 ⊢ (𝑥(+g‘𝑂)(𝑧(+g‘𝑅)𝑦)) = ((𝑧(+g‘𝑅)𝑦)(+g‘𝑅)𝑥) |
25 | 23, 24 | eqtri 2632 | . . 3 ⊢ (𝑥(+g‘𝑂)(𝑦(+g‘𝑂)𝑧)) = ((𝑧(+g‘𝑅)𝑦)(+g‘𝑅)𝑥) |
26 | 18, 21, 25 | 3eqtr4g 2669 | . 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 2610 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
28 | 2, 27 | mndidcl 17131 | . 2 ⊢ (𝑅 ∈ Mnd → (0g‘𝑅) ∈ (Base‘𝑅)) |
29 | 6, 1, 7 | oppgplus 17602 | . . 3 ⊢ ((0g‘𝑅)(+g‘𝑂)𝑥) = (𝑥(+g‘𝑅)(0g‘𝑅)) |
30 | 2, 6, 27 | mndrid 17135 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(0g‘𝑅)) = 𝑥) |
31 | 29, 30 | syl5eq 2656 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑂)𝑥) = 𝑥) |
32 | 6, 1, 7 | oppgplus 17602 | . . 3 ⊢ (𝑥(+g‘𝑂)(0g‘𝑅)) = ((0g‘𝑅)(+g‘𝑅)𝑥) |
33 | 2, 6, 27 | mndlid 17134 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑥) = 𝑥) |
34 | 32, 33 | syl5eq 2656 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑂)(0g‘𝑅)) = 𝑥) |
35 | 4, 5, 11, 26, 28, 31, 34 | ismndd 17136 | 1 ⊢ (𝑅 ∈ Mnd → 𝑂 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Mndcmnd 17117 oppgcoppg 17598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-oppg 17599 |
This theorem is referenced by: oppgmndb 17608 oppggrp 17610 gsumwrev 17619 gsumzoppg 18167 gsumzinv 18168 oppgtmd 21711 |
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