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Mirrors > Home > MPE Home > Th. List > gsumzsubmcl | Structured version Visualization version GIF version |
Description: Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.) |
Ref | Expression |
---|---|
gsumzsubmcl.0 | ⊢ 0 = (0g‘𝐺) |
gsumzsubmcl.z | ⊢ 𝑍 = (Cntz‘𝐺) |
gsumzsubmcl.g | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
gsumzsubmcl.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumzsubmcl.s | ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) |
gsumzsubmcl.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
gsumzsubmcl.c | ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
gsumzsubmcl.w | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Ref | Expression |
---|---|
gsumzsubmcl | ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ (Base‘(𝐺 ↾s 𝑆)) = (Base‘(𝐺 ↾s 𝑆)) | |
2 | eqid 2798 | . . 3 ⊢ (0g‘(𝐺 ↾s 𝑆)) = (0g‘(𝐺 ↾s 𝑆)) | |
3 | eqid 2798 | . . 3 ⊢ (Cntz‘(𝐺 ↾s 𝑆)) = (Cntz‘(𝐺 ↾s 𝑆)) | |
4 | gsumzsubmcl.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) | |
5 | eqid 2798 | . . . . 5 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆) | |
6 | 5 | submmnd 17970 | . . . 4 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (𝐺 ↾s 𝑆) ∈ Mnd) |
7 | 4, 6 | syl 17 | . . 3 ⊢ (𝜑 → (𝐺 ↾s 𝑆) ∈ Mnd) |
8 | gsumzsubmcl.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
9 | gsumzsubmcl.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
10 | 5 | submbas 17971 | . . . . . 6 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
11 | 4, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
12 | 11 | feq3d 6474 | . . . 4 ⊢ (𝜑 → (𝐹:𝐴⟶𝑆 ↔ 𝐹:𝐴⟶(Base‘(𝐺 ↾s 𝑆)))) |
13 | 9, 12 | mpbid 235 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘(𝐺 ↾s 𝑆))) |
14 | gsumzsubmcl.c | . . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) | |
15 | 9 | frnd 6494 | . . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑆) |
16 | 14, 15 | ssind 4159 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ((𝑍‘ran 𝐹) ∩ 𝑆)) |
17 | gsumzsubmcl.z | . . . . . 6 ⊢ 𝑍 = (Cntz‘𝐺) | |
18 | 5, 17, 3 | resscntz 18454 | . . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ ran 𝐹 ⊆ 𝑆) → ((Cntz‘(𝐺 ↾s 𝑆))‘ran 𝐹) = ((𝑍‘ran 𝐹) ∩ 𝑆)) |
19 | 4, 15, 18 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ((Cntz‘(𝐺 ↾s 𝑆))‘ran 𝐹) = ((𝑍‘ran 𝐹) ∩ 𝑆)) |
20 | 16, 19 | sseqtrrd 3956 | . . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ((Cntz‘(𝐺 ↾s 𝑆))‘ran 𝐹)) |
21 | gsumzsubmcl.w | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
22 | gsumzsubmcl.0 | . . . . . 6 ⊢ 0 = (0g‘𝐺) | |
23 | 5, 22 | subm0 17972 | . . . . 5 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 0 = (0g‘(𝐺 ↾s 𝑆))) |
24 | 4, 23 | syl 17 | . . . 4 ⊢ (𝜑 → 0 = (0g‘(𝐺 ↾s 𝑆))) |
25 | 21, 24 | breqtrd 5056 | . . 3 ⊢ (𝜑 → 𝐹 finSupp (0g‘(𝐺 ↾s 𝑆))) |
26 | 1, 2, 3, 7, 8, 13, 20, 25 | gsumzcl 19024 | . 2 ⊢ (𝜑 → ((𝐺 ↾s 𝑆) Σg 𝐹) ∈ (Base‘(𝐺 ↾s 𝑆))) |
27 | 8, 4, 9, 5 | gsumsubm 17991 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 ↾s 𝑆) Σg 𝐹)) |
28 | 26, 27, 11 | 3eltr4d 2905 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 class class class wbr 5030 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 finSupp cfsupp 8817 Basecbs 16475 ↾s cress 16476 0gc0g 16705 Σg cgsu 16706 Mndcmnd 17903 SubMndcsubmnd 17947 Cntzccntz 18437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-gsum 16708 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-cntz 18439 |
This theorem is referenced by: gsumsubmcl 19032 gsumzadd 19035 dprdfadd 19135 dprdfeq0 19137 dprdlub 19141 |
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