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Mirrors > Home > MPE Home > Th. List > oppgsubm | Structured version Visualization version GIF version |
Description: Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.) |
Ref | Expression |
---|---|
oppggic.o | ⊢ 𝑂 = (oppg‘𝐺) |
Ref | Expression |
---|---|
oppgsubm | ⊢ (SubMnd‘𝐺) = (SubMnd‘𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | submrcl 18717 | . . 3 ⊢ (𝑥 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd) | |
2 | submrcl 18717 | . . . 4 ⊢ (𝑥 ∈ (SubMnd‘𝑂) → 𝑂 ∈ Mnd) | |
3 | oppggic.o | . . . . 5 ⊢ 𝑂 = (oppg‘𝐺) | |
4 | 3 | oppgmndb 19264 | . . . 4 ⊢ (𝐺 ∈ Mnd ↔ 𝑂 ∈ Mnd) |
5 | 2, 4 | sylibr 233 | . . 3 ⊢ (𝑥 ∈ (SubMnd‘𝑂) → 𝐺 ∈ Mnd) |
6 | ralcom 3278 | . . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝐺)𝑧) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑦(+g‘𝐺)𝑧) ∈ 𝑥) | |
7 | eqid 2724 | . . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
8 | eqid 2724 | . . . . . . . . . 10 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
9 | 7, 3, 8 | oppgplus 19255 | . . . . . . . . 9 ⊢ (𝑧(+g‘𝑂)𝑦) = (𝑦(+g‘𝐺)𝑧) |
10 | 9 | eleq1i 2816 | . . . . . . . 8 ⊢ ((𝑧(+g‘𝑂)𝑦) ∈ 𝑥 ↔ (𝑦(+g‘𝐺)𝑧) ∈ 𝑥) |
11 | 10 | 2ralbii 3120 | . . . . . . 7 ⊢ (∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(+g‘𝑂)𝑦) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑦(+g‘𝐺)𝑧) ∈ 𝑥) |
12 | 6, 11 | bitr4i 278 | . . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝐺)𝑧) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(+g‘𝑂)𝑦) ∈ 𝑥) |
13 | 12 | 3anbi3i 1156 | . . . . 5 ⊢ ((𝑥 ⊆ (Base‘𝐺) ∧ (0g‘𝐺) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝐺)𝑧) ∈ 𝑥) ↔ (𝑥 ⊆ (Base‘𝐺) ∧ (0g‘𝐺) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(+g‘𝑂)𝑦) ∈ 𝑥)) |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝐺 ∈ Mnd → ((𝑥 ⊆ (Base‘𝐺) ∧ (0g‘𝐺) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝐺)𝑧) ∈ 𝑥) ↔ (𝑥 ⊆ (Base‘𝐺) ∧ (0g‘𝐺) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(+g‘𝑂)𝑦) ∈ 𝑥))) |
15 | eqid 2724 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
16 | eqid 2724 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
17 | 15, 16, 7 | issubm 18718 | . . . 4 ⊢ (𝐺 ∈ Mnd → (𝑥 ∈ (SubMnd‘𝐺) ↔ (𝑥 ⊆ (Base‘𝐺) ∧ (0g‘𝐺) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝐺)𝑧) ∈ 𝑥))) |
18 | 3, 15 | oppgbas 19258 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝑂) |
19 | 3, 16 | oppgid 19265 | . . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝑂) |
20 | 18, 19, 8 | issubm 18718 | . . . . 5 ⊢ (𝑂 ∈ Mnd → (𝑥 ∈ (SubMnd‘𝑂) ↔ (𝑥 ⊆ (Base‘𝐺) ∧ (0g‘𝐺) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(+g‘𝑂)𝑦) ∈ 𝑥))) |
21 | 4, 20 | sylbi 216 | . . . 4 ⊢ (𝐺 ∈ Mnd → (𝑥 ∈ (SubMnd‘𝑂) ↔ (𝑥 ⊆ (Base‘𝐺) ∧ (0g‘𝐺) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(+g‘𝑂)𝑦) ∈ 𝑥))) |
22 | 14, 17, 21 | 3bitr4d 311 | . . 3 ⊢ (𝐺 ∈ Mnd → (𝑥 ∈ (SubMnd‘𝐺) ↔ 𝑥 ∈ (SubMnd‘𝑂))) |
23 | 1, 5, 22 | pm5.21nii 378 | . 2 ⊢ (𝑥 ∈ (SubMnd‘𝐺) ↔ 𝑥 ∈ (SubMnd‘𝑂)) |
24 | 23 | eqriv 2721 | 1 ⊢ (SubMnd‘𝐺) = (SubMnd‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ⊆ wss 3940 ‘cfv 6533 (class class class)co 7401 Basecbs 17143 +gcplusg 17196 0gc0g 17384 Mndcmnd 18657 SubMndcsubmnd 18702 oppgcoppg 19251 |
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 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-0g 17386 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-oppg 19252 |
This theorem is referenced by: oppgsubg 19272 gsumzoppg 19854 |
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