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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 2821 | . . 3 ⊢ (Base‘(𝐺 ↾s 𝑆)) = (Base‘(𝐺 ↾s 𝑆)) | |
2 | eqid 2821 | . . 3 ⊢ (0g‘(𝐺 ↾s 𝑆)) = (0g‘(𝐺 ↾s 𝑆)) | |
3 | eqid 2821 | . . 3 ⊢ (Cntz‘(𝐺 ↾s 𝑆)) = (Cntz‘(𝐺 ↾s 𝑆)) | |
4 | gsumzsubmcl.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) | |
5 | eqid 2821 | . . . . 5 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆) | |
6 | 5 | submmnd 17972 | . . . 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 17973 | . . . . . 6 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
11 | 4, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
12 | 11 | feq3d 6495 | . . . 4 ⊢ (𝜑 → (𝐹:𝐴⟶𝑆 ↔ 𝐹:𝐴⟶(Base‘(𝐺 ↾s 𝑆)))) |
13 | 9, 12 | mpbid 234 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘(𝐺 ↾s 𝑆))) |
14 | gsumzsubmcl.c | . . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) | |
15 | 9 | frnd 6515 | . . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑆) |
16 | 14, 15 | ssind 4208 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ((𝑍‘ran 𝐹) ∩ 𝑆)) |
17 | gsumzsubmcl.z | . . . . . 6 ⊢ 𝑍 = (Cntz‘𝐺) | |
18 | 5, 17, 3 | resscntz 18456 | . . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ ran 𝐹 ⊆ 𝑆) → ((Cntz‘(𝐺 ↾s 𝑆))‘ran 𝐹) = ((𝑍‘ran 𝐹) ∩ 𝑆)) |
19 | 4, 15, 18 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ((Cntz‘(𝐺 ↾s 𝑆))‘ran 𝐹) = ((𝑍‘ran 𝐹) ∩ 𝑆)) |
20 | 16, 19 | sseqtrrd 4007 | . . 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 17974 | . . . . 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 5084 | . . 3 ⊢ (𝜑 → 𝐹 finSupp (0g‘(𝐺 ↾s 𝑆))) |
26 | 1, 2, 3, 7, 8, 13, 20, 25 | gsumzcl 19025 | . 2 ⊢ (𝜑 → ((𝐺 ↾s 𝑆) Σg 𝐹) ∈ (Base‘(𝐺 ↾s 𝑆))) |
27 | 8, 4, 9, 5 | gsumsubm 17993 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 ↾s 𝑆) Σg 𝐹)) |
28 | 26, 27, 11 | 3eltr4d 2928 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ⊆ wss 3935 class class class wbr 5058 ran crn 5550 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 finSupp cfsupp 8827 Basecbs 16477 ↾s cress 16478 0gc0g 16707 Σg cgsu 16708 Mndcmnd 17905 SubMndcsubmnd 17949 Cntzccntz 18439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-0g 16709 df-gsum 16710 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-cntz 18441 |
This theorem is referenced by: gsumsubmcl 19033 gsumzadd 19036 dprdfadd 19136 dprdfeq0 19138 dprdlub 19142 |
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