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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 2738 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → (Base‘𝐻) = (Base‘𝐻)) | |
| 2 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
| 3 | eqid 2737 | . . . . . . 7 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
| 4 | 2, 3 | mndidcl 18762 | . . . . . 6 ⊢ (𝐻 ∈ Mnd → (0g‘𝐻) ∈ (Base‘𝐻)) |
| 5 | n0i 4340 | . . . . . 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 17276 | . . . . . . . . 9 ⊢ Rel dom ↾s | |
| 9 | 8 | ovprc2 7471 | . . . . . . . 8 ⊢ (¬ 𝑆 ∈ V → (𝐺 ↾s 𝑆) = ∅) |
| 10 | 7, 9 | eqtrid 2789 | . . . . . . 7 ⊢ (¬ 𝑆 ∈ V → 𝐻 = ∅) |
| 11 | 10 | fveq2d 6910 | . . . . . 6 ⊢ (¬ 𝑆 ∈ V → (Base‘𝐻) = (Base‘∅)) |
| 12 | base0 17252 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
| 13 | 11, 12 | eqtr4di 2795 | . . . . 5 ⊢ (¬ 𝑆 ∈ V → (Base‘𝐻) = ∅) |
| 14 | 6, 13 | nsyl2 141 | . . . 4 ⊢ (𝐻 ∈ Mnd → 𝑆 ∈ V) |
| 15 | 14 | adantl 481 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝑆 ∈ V) |
| 16 | eqid 2737 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 17 | 7, 16 | ressplusg 17334 | . . 3 ⊢ (𝑆 ∈ V → (+g‘𝐺) = (+g‘𝐻)) |
| 18 | 15, 17 | syl 17 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → (+g‘𝐺) = (+g‘𝐻)) |
| 19 | simpr 484 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝐻 ∈ Mnd) | |
| 20 | simpl 482 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝐺 ∈ CMnd) | |
| 21 | eqid 2737 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 22 | 7, 21 | ressbasss 17284 | . . . 4 ⊢ (Base‘𝐻) ⊆ (Base‘𝐺) |
| 23 | 22 | sseli 3979 | . . 3 ⊢ (𝑥 ∈ (Base‘𝐻) → 𝑥 ∈ (Base‘𝐺)) |
| 24 | 22 | sseli 3979 | . . 3 ⊢ (𝑦 ∈ (Base‘𝐻) → 𝑦 ∈ (Base‘𝐺)) |
| 25 | 21, 16 | cmncom 19816 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) |
| 26 | 20, 23, 24, 25 | syl3an 1161 | . 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 19812 | 1 ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝐻 ∈ CMnd) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 +gcplusg 17297 0gc0g 17484 Mndcmnd 18747 CMndccmn 19798 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-cmn 19800 |
| This theorem is referenced by: submcmn 19856 unitabl 20384 subrgcrng 20575 xrge0cmn 21426 tsmssubm 24151 amgmlem 27033 amgmwlem 49321 amgmlemALT 49322 |
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