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Mirrors > Home > MPE Home > Th. List > gsumsubmcl | Structured version Visualization version GIF version |
Description: Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.) |
Ref | Expression |
---|---|
gsumsubmcl.z | ⊢ 0 = (0g‘𝐺) |
gsumsubmcl.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumsubmcl.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumsubmcl.s | ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) |
gsumsubmcl.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
gsumsubmcl.w | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Ref | Expression |
---|---|
gsumsubmcl | ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumsubmcl.z | . 2 ⊢ 0 = (0g‘𝐺) | |
2 | eqid 2740 | . 2 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
3 | gsumsubmcl.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | cmnmnd 19413 | . . 3 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
6 | gsumsubmcl.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | gsumsubmcl.s | . 2 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) | |
8 | gsumsubmcl.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
9 | eqid 2740 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
10 | 9 | submss 18459 | . . . . 5 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
11 | 7, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐺)) |
12 | 8, 11 | fssd 6616 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘𝐺)) |
13 | 9, 2, 3, 12 | cntzcmnf 19457 | . 2 ⊢ (𝜑 → ran 𝐹 ⊆ ((Cntz‘𝐺)‘ran 𝐹)) |
14 | gsumsubmcl.w | . 2 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
15 | 1, 2, 5, 6, 7, 8, 13, 14 | gsumzsubmcl 19530 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ⊆ wss 3892 class class class wbr 5079 ⟶wf 6428 ‘cfv 6432 (class class class)co 7272 finSupp cfsupp 9116 Basecbs 16923 0gc0g 17161 Σg cgsu 17162 Mndcmnd 18396 SubMndcsubmnd 18440 Cntzccntz 18932 CMndccmn 19397 |
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 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-supp 7970 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-fsupp 9117 df-oi 9257 df-card 9708 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-2 12047 df-n0 12245 df-z 12331 df-uz 12594 df-fz 13251 df-fzo 13394 df-seq 13733 df-hash 14056 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-ress 16953 df-plusg 16986 df-0g 17163 df-gsum 17164 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-submnd 18442 df-cntz 18934 df-cmn 19399 |
This theorem is referenced by: gsumsubgcl 19532 mplbas2 21254 tdeglem1 25231 tdeglem1OLD 25232 tdeglem4 25235 tdeglem4OLD 25236 plypf1 25384 jensen 26149 amgmlem 26150 amgm 26151 wilthlem2 26229 wilthlem3 26230 lgseisenlem3 26536 elrspunidl 31615 amgmwlem 46485 |
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