![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > subcmnd | GIF version |
Description: A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
subcmnd.h | ⊢ (𝜑 → 𝐻 = (𝐺 ↾s 𝑆)) |
subcmnd.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
subcmnd.m | ⊢ (𝜑 → 𝐻 ∈ Mnd) |
subcmnd.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
Ref | Expression |
---|---|
subcmnd | ⊢ (𝜑 → 𝐻 ∈ CMnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2178 | . 2 ⊢ (𝜑 → (Base‘𝐻) = (Base‘𝐻)) | |
2 | subcmnd.h | . . 3 ⊢ (𝜑 → 𝐻 = (𝐺 ↾s 𝑆)) | |
3 | eqidd 2178 | . . 3 ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐺)) | |
4 | subcmnd.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
5 | subcmnd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
6 | 2, 3, 4, 5 | ressplusgd 12581 | . 2 ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻)) |
7 | subcmnd.m | . 2 ⊢ (𝜑 → 𝐻 ∈ Mnd) | |
8 | 5 | 3ad2ant1 1018 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐻) ∧ 𝑦 ∈ (Base‘𝐻)) → 𝐺 ∈ CMnd) |
9 | eqidd 2178 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺)) | |
10 | 2, 9, 5, 4 | ressbasssd 12523 | . . . . 5 ⊢ (𝜑 → (Base‘𝐻) ⊆ (Base‘𝐺)) |
11 | 10 | sselda 3155 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐻)) → 𝑥 ∈ (Base‘𝐺)) |
12 | 11 | 3adant3 1017 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐻) ∧ 𝑦 ∈ (Base‘𝐻)) → 𝑥 ∈ (Base‘𝐺)) |
13 | 10 | sselda 3155 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐻)) → 𝑦 ∈ (Base‘𝐺)) |
14 | 13 | 3adant2 1016 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐻) ∧ 𝑦 ∈ (Base‘𝐻)) → 𝑦 ∈ (Base‘𝐺)) |
15 | eqid 2177 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
16 | eqid 2177 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
17 | 15, 16 | cmncom 13058 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) |
18 | 8, 12, 14, 17 | syl3anc 1238 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐻) ∧ 𝑦 ∈ (Base‘𝐻)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) |
19 | 1, 6, 7, 18 | iscmnd 13054 | 1 ⊢ (𝜑 → 𝐻 ∈ CMnd) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ‘cfv 5216 (class class class)co 5874 Basecbs 12456 ↾s cress 12457 +gcplusg 12530 Mndcmnd 12771 CMndccmn 13041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fun 5218 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-inn 8918 df-2 8976 df-ndx 12459 df-slot 12460 df-base 12462 df-sets 12463 df-iress 12464 df-plusg 12543 df-cmn 13043 |
This theorem is referenced by: unitabl 13239 subrgcrng 13306 |
Copyright terms: Public domain | W3C validator |