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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 2737 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 2 | 1 | submacs 18840 | . . . . 5 ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘(Base‘𝐺))) |
| 3 | 2 | adantr 480 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (SubMnd‘𝐺) ∈ (ACS‘(Base‘𝐺))) |
| 4 | 3 | acsmred 17699 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (SubMnd‘𝐺) ∈ (Moore‘(Base‘𝐺))) |
| 5 | simpr 484 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → 𝑆 ⊆ (𝑍‘𝑆)) | |
| 6 | cntzspan.z | . . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝐺) | |
| 7 | 1, 6 | cntzssv 19346 | . . . . . . 7 ⊢ (𝑍‘𝑆) ⊆ (Base‘𝐺) |
| 8 | 5, 7 | sstrdi 3996 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → 𝑆 ⊆ (Base‘𝐺)) |
| 9 | 1, 6 | cntzsubm 19356 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (Base‘𝐺)) → (𝑍‘𝑆) ∈ (SubMnd‘𝐺)) |
| 10 | 8, 9 | syldan 591 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (𝑍‘𝑆) ∈ (SubMnd‘𝐺)) |
| 11 | cntzspan.k | . . . . . 6 ⊢ 𝐾 = (mrCls‘(SubMnd‘𝐺)) | |
| 12 | 11 | mrcsscl 17663 | . . . . 5 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ 𝑆 ⊆ (𝑍‘𝑆) ∧ (𝑍‘𝑆) ∈ (SubMnd‘𝐺)) → (𝐾‘𝑆) ⊆ (𝑍‘𝑆)) |
| 13 | 4, 5, 10, 12 | syl3anc 1373 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (𝐾‘𝑆) ⊆ (𝑍‘𝑆)) |
| 14 | 4, 11 | mrcssvd 17666 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (𝐾‘𝑆) ⊆ (Base‘𝐺)) |
| 15 | 1, 6 | cntzrec 19354 | . . . . 5 ⊢ (((𝐾‘𝑆) ⊆ (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺)) → ((𝐾‘𝑆) ⊆ (𝑍‘𝑆) ↔ 𝑆 ⊆ (𝑍‘(𝐾‘𝑆)))) |
| 16 | 14, 8, 15 | syl2anc 584 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → ((𝐾‘𝑆) ⊆ (𝑍‘𝑆) ↔ 𝑆 ⊆ (𝑍‘(𝐾‘𝑆)))) |
| 17 | 13, 16 | mpbid 232 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → 𝑆 ⊆ (𝑍‘(𝐾‘𝑆))) |
| 18 | 1, 6 | cntzsubm 19356 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ (𝐾‘𝑆) ⊆ (Base‘𝐺)) → (𝑍‘(𝐾‘𝑆)) ∈ (SubMnd‘𝐺)) |
| 19 | 14, 18 | syldan 591 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (𝑍‘(𝐾‘𝑆)) ∈ (SubMnd‘𝐺)) |
| 20 | 11 | mrcsscl 17663 | . . 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 1373 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (𝐾‘𝑆) ⊆ (𝑍‘(𝐾‘𝑆))) |
| 22 | 11 | mrccl 17654 | . . . 4 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ 𝑆 ⊆ (Base‘𝐺)) → (𝐾‘𝑆) ∈ (SubMnd‘𝐺)) |
| 23 | 4, 8, 22 | syl2anc 584 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → (𝐾‘𝑆) ∈ (SubMnd‘𝐺)) |
| 24 | cntzspan.h | . . . 4 ⊢ 𝐻 = (𝐺 ↾s (𝐾‘𝑆)) | |
| 25 | 24, 6 | submcmn2 19857 | . . 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 257 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → 𝐻 ∈ CMnd) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 Moorecmre 17625 mrClscmrc 17626 ACScacs 17628 Mndcmnd 18747 SubMndcsubmnd 18795 Cntzccntz 19333 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-rep 5279 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-int 4947 df-iun 4993 df-iin 4994 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-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 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-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-cntz 19335 df-cmn 19800 |
| This theorem is referenced by: gsumzsplit 19945 gsumzoppg 19962 gsumpt 19980 |
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