![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > subcmn | Structured version Visualization version GIF version |
Description: A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
subgabl.h | ⊢ 𝐻 = (𝐺 ↾s 𝑆) |
Ref | Expression |
---|---|
subcmn | ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝐻 ∈ CMnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2732 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → (Base‘𝐻) = (Base‘𝐻)) | |
2 | eqid 2731 | . . . . . . 7 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
3 | eqid 2731 | . . . . . . 7 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
4 | 2, 3 | mndidcl 18617 | . . . . . 6 ⊢ (𝐻 ∈ Mnd → (0g‘𝐻) ∈ (Base‘𝐻)) |
5 | n0i 4329 | . . . . . 6 ⊢ ((0g‘𝐻) ∈ (Base‘𝐻) → ¬ (Base‘𝐻) = ∅) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝐻 ∈ Mnd → ¬ (Base‘𝐻) = ∅) |
7 | subgabl.h | . . . . . . . 8 ⊢ 𝐻 = (𝐺 ↾s 𝑆) | |
8 | reldmress 17157 | . . . . . . . . 9 ⊢ Rel dom ↾s | |
9 | 8 | ovprc2 7433 | . . . . . . . 8 ⊢ (¬ 𝑆 ∈ V → (𝐺 ↾s 𝑆) = ∅) |
10 | 7, 9 | eqtrid 2783 | . . . . . . 7 ⊢ (¬ 𝑆 ∈ V → 𝐻 = ∅) |
11 | 10 | fveq2d 6882 | . . . . . 6 ⊢ (¬ 𝑆 ∈ V → (Base‘𝐻) = (Base‘∅)) |
12 | base0 17131 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
13 | 11, 12 | eqtr4di 2789 | . . . . 5 ⊢ (¬ 𝑆 ∈ V → (Base‘𝐻) = ∅) |
14 | 6, 13 | nsyl2 141 | . . . 4 ⊢ (𝐻 ∈ Mnd → 𝑆 ∈ V) |
15 | 14 | adantl 482 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝑆 ∈ V) |
16 | eqid 2731 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
17 | 7, 16 | ressplusg 17217 | . . 3 ⊢ (𝑆 ∈ V → (+g‘𝐺) = (+g‘𝐻)) |
18 | 15, 17 | syl 17 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → (+g‘𝐺) = (+g‘𝐻)) |
19 | simpr 485 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝐻 ∈ Mnd) | |
20 | simpl 483 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝐺 ∈ CMnd) | |
21 | eqid 2731 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
22 | 7, 21 | ressbasss 17165 | . . . 4 ⊢ (Base‘𝐻) ⊆ (Base‘𝐺) |
23 | 22 | sseli 3974 | . . 3 ⊢ (𝑥 ∈ (Base‘𝐻) → 𝑥 ∈ (Base‘𝐺)) |
24 | 22 | sseli 3974 | . . 3 ⊢ (𝑦 ∈ (Base‘𝐻) → 𝑦 ∈ (Base‘𝐺)) |
25 | 21, 16 | cmncom 19630 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) |
26 | 20, 23, 24, 25 | syl3an 1160 | . 2 ⊢ (((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) ∧ 𝑥 ∈ (Base‘𝐻) ∧ 𝑦 ∈ (Base‘𝐻)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) |
27 | 1, 18, 19, 26 | iscmnd 19626 | 1 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝐻 ∈ CMnd) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3473 ∅c0 4318 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 ↾s cress 17155 +gcplusg 17179 0gc0g 17367 Mndcmnd 18602 CMndccmn 19612 |
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-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-ress 17156 df-plusg 17192 df-0g 17369 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-cmn 19614 |
This theorem is referenced by: submcmn 19666 unitabl 20150 subrgcrng 20316 xrge0cmn 20921 tsmssubm 23576 amgmlem 26421 amgmwlem 47497 amgmlemALT 47498 |
Copyright terms: Public domain | W3C validator |