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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 2822 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → (Base‘𝐻) = (Base‘𝐻)) | |
2 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
3 | eqid 2821 | . . . . . . 7 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
4 | 2, 3 | mndidcl 17920 | . . . . . 6 ⊢ (𝐻 ∈ Mnd → (0g‘𝐻) ∈ (Base‘𝐻)) |
5 | n0i 4298 | . . . . . 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 16544 | . . . . . . . . 9 ⊢ Rel dom ↾s | |
9 | 8 | ovprc2 7190 | . . . . . . . 8 ⊢ (¬ 𝑆 ∈ V → (𝐺 ↾s 𝑆) = ∅) |
10 | 7, 9 | syl5eq 2868 | . . . . . . 7 ⊢ (¬ 𝑆 ∈ V → 𝐻 = ∅) |
11 | 10 | fveq2d 6668 | . . . . . 6 ⊢ (¬ 𝑆 ∈ V → (Base‘𝐻) = (Base‘∅)) |
12 | base0 16530 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
13 | 11, 12 | syl6eqr 2874 | . . . . 5 ⊢ (¬ 𝑆 ∈ V → (Base‘𝐻) = ∅) |
14 | 6, 13 | nsyl2 143 | . . . 4 ⊢ (𝐻 ∈ Mnd → 𝑆 ∈ V) |
15 | 14 | adantl 484 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝑆 ∈ V) |
16 | eqid 2821 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
17 | 7, 16 | ressplusg 16606 | . . 3 ⊢ (𝑆 ∈ V → (+g‘𝐺) = (+g‘𝐻)) |
18 | 15, 17 | syl 17 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → (+g‘𝐺) = (+g‘𝐻)) |
19 | simpr 487 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝐻 ∈ Mnd) | |
20 | simpl 485 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝐺 ∈ CMnd) | |
21 | eqid 2821 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
22 | 7, 21 | ressbasss 16550 | . . . 4 ⊢ (Base‘𝐻) ⊆ (Base‘𝐺) |
23 | 22 | sseli 3962 | . . 3 ⊢ (𝑥 ∈ (Base‘𝐻) → 𝑥 ∈ (Base‘𝐺)) |
24 | 22 | sseli 3962 | . . 3 ⊢ (𝑦 ∈ (Base‘𝐻) → 𝑦 ∈ (Base‘𝐺)) |
25 | 21, 16 | cmncom 18917 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) |
26 | 20, 23, 24, 25 | syl3an 1156 | . 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 18913 | 1 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝐻 ∈ CMnd) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 ↾s cress 16478 +gcplusg 16559 0gc0g 16707 Mndcmnd 17905 CMndccmn 18900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-cmn 18902 |
This theorem is referenced by: submcmn 18952 unitabl 19412 subrgcrng 19533 xrge0cmn 20581 tsmssubm 22745 amgmlem 25561 amgmwlem 44897 amgmlemALT 44898 |
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