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Mirrors > Home > MPE Home > Th. List > cntzspan | Structured version Visualization version GIF version |
Description: If the generators commute, the generated monoid is commutative. (Contributed by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
cntzspan.z | ⊢ 𝑍 = (Cntz‘𝐺) |
cntzspan.k | ⊢ 𝐾 = (mrCls‘(SubMnd‘𝐺)) |
cntzspan.h | ⊢ 𝐻 = (𝐺 ↾s (𝐾‘𝑆)) |
Ref | Expression |
---|---|
cntzspan | ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → 𝐻 ∈ CMnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 1 | submacs 18107 | . . . . 5 ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘(Base‘𝐺))) |
3 | 2 | adantr 484 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (SubMnd‘𝐺) ∈ (ACS‘(Base‘𝐺))) |
4 | 3 | acsmred 17030 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (SubMnd‘𝐺) ∈ (Moore‘(Base‘𝐺))) |
5 | simpr 488 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → 𝑆 ⊆ (𝑍‘𝑆)) | |
6 | cntzspan.z | . . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝐺) | |
7 | 1, 6 | cntzssv 18576 | . . . . . . 7 ⊢ (𝑍‘𝑆) ⊆ (Base‘𝐺) |
8 | 5, 7 | sstrdi 3889 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → 𝑆 ⊆ (Base‘𝐺)) |
9 | 1, 6 | cntzsubm 18584 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (Base‘𝐺)) → (𝑍‘𝑆) ∈ (SubMnd‘𝐺)) |
10 | 8, 9 | syldan 594 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (𝑍‘𝑆) ∈ (SubMnd‘𝐺)) |
11 | cntzspan.k | . . . . . 6 ⊢ 𝐾 = (mrCls‘(SubMnd‘𝐺)) | |
12 | 11 | mrcsscl 16994 | . . . . 5 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ 𝑆 ⊆ (𝑍‘𝑆) ∧ (𝑍‘𝑆) ∈ (SubMnd‘𝐺)) → (𝐾‘𝑆) ⊆ (𝑍‘𝑆)) |
13 | 4, 5, 10, 12 | syl3anc 1372 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (𝐾‘𝑆) ⊆ (𝑍‘𝑆)) |
14 | 4, 11 | mrcssvd 16997 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (𝐾‘𝑆) ⊆ (Base‘𝐺)) |
15 | 1, 6 | cntzrec 18582 | . . . . 5 ⊢ (((𝐾‘𝑆) ⊆ (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺)) → ((𝐾‘𝑆) ⊆ (𝑍‘𝑆) ↔ 𝑆 ⊆ (𝑍‘(𝐾‘𝑆)))) |
16 | 14, 8, 15 | syl2anc 587 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → ((𝐾‘𝑆) ⊆ (𝑍‘𝑆) ↔ 𝑆 ⊆ (𝑍‘(𝐾‘𝑆)))) |
17 | 13, 16 | mpbid 235 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → 𝑆 ⊆ (𝑍‘(𝐾‘𝑆))) |
18 | 1, 6 | cntzsubm 18584 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ (𝐾‘𝑆) ⊆ (Base‘𝐺)) → (𝑍‘(𝐾‘𝑆)) ∈ (SubMnd‘𝐺)) |
19 | 14, 18 | syldan 594 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (𝑍‘(𝐾‘𝑆)) ∈ (SubMnd‘𝐺)) |
20 | 11 | mrcsscl 16994 | . . 3 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ 𝑆 ⊆ (𝑍‘(𝐾‘𝑆)) ∧ (𝑍‘(𝐾‘𝑆)) ∈ (SubMnd‘𝐺)) → (𝐾‘𝑆) ⊆ (𝑍‘(𝐾‘𝑆))) |
21 | 4, 17, 19, 20 | syl3anc 1372 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (𝐾‘𝑆) ⊆ (𝑍‘(𝐾‘𝑆))) |
22 | 11 | mrccl 16985 | . . . 4 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ 𝑆 ⊆ (Base‘𝐺)) → (𝐾‘𝑆) ∈ (SubMnd‘𝐺)) |
23 | 4, 8, 22 | syl2anc 587 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (𝐾‘𝑆) ∈ (SubMnd‘𝐺)) |
24 | cntzspan.h | . . . 4 ⊢ 𝐻 = (𝐺 ↾s (𝐾‘𝑆)) | |
25 | 24, 6 | submcmn2 19078 | . . 3 ⊢ ((𝐾‘𝑆) ∈ (SubMnd‘𝐺) → (𝐻 ∈ CMnd ↔ (𝐾‘𝑆) ⊆ (𝑍‘(𝐾‘𝑆)))) |
26 | 23, 25 | syl 17 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (𝐻 ∈ CMnd ↔ (𝐾‘𝑆) ⊆ (𝑍‘(𝐾‘𝑆)))) |
27 | 21, 26 | mpbird 260 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → 𝐻 ∈ CMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ⊆ wss 3843 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 ↾s cress 16587 Moorecmre 16956 mrClscmrc 16957 ACScacs 16959 Mndcmnd 18027 SubMndcsubmnd 18071 Cntzccntz 18563 CMndccmn 19024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-0g 16818 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-cntz 18565 df-cmn 19026 |
This theorem is referenced by: gsumzsplit 19166 gsumzoppg 19183 gsumpt 19201 |
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