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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 2823 | . . 3 ⊢ (Base‘(𝐺 ↾s 𝑆)) = (Base‘(𝐺 ↾s 𝑆)) | |
2 | eqid 2823 | . . 3 ⊢ (0g‘(𝐺 ↾s 𝑆)) = (0g‘(𝐺 ↾s 𝑆)) | |
3 | eqid 2823 | . . 3 ⊢ (Cntz‘(𝐺 ↾s 𝑆)) = (Cntz‘(𝐺 ↾s 𝑆)) | |
4 | gsumzsubmcl.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) | |
5 | eqid 2823 | . . . . 5 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆) | |
6 | 5 | submmnd 17980 | . . . 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 17981 | . . . . . 6 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
11 | 4, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
12 | 11 | feq3d 6503 | . . . 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 6523 | . . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑆) |
16 | 14, 15 | ssind 4211 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ((𝑍‘ran 𝐹) ∩ 𝑆)) |
17 | gsumzsubmcl.z | . . . . . 6 ⊢ 𝑍 = (Cntz‘𝐺) | |
18 | 5, 17, 3 | resscntz 18464 | . . . . 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 4010 | . . 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 17982 | . . . . 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 5094 | . . 3 ⊢ (𝜑 → 𝐹 finSupp (0g‘(𝐺 ↾s 𝑆))) |
26 | 1, 2, 3, 7, 8, 13, 20, 25 | gsumzcl 19033 | . 2 ⊢ (𝜑 → ((𝐺 ↾s 𝑆) Σg 𝐹) ∈ (Base‘(𝐺 ↾s 𝑆))) |
27 | 8, 4, 9, 5 | gsumsubm 18001 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 ↾s 𝑆) Σg 𝐹)) |
28 | 26, 27, 11 | 3eltr4d 2930 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ⊆ wss 3938 class class class wbr 5068 ran crn 5558 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 finSupp cfsupp 8835 Basecbs 16485 ↾s cress 16486 0gc0g 16715 Σg cgsu 16716 Mndcmnd 17913 SubMndcsubmnd 17957 Cntzccntz 18447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-0g 16717 df-gsum 16718 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-cntz 18449 |
This theorem is referenced by: gsumsubmcl 19041 gsumzadd 19044 dprdfadd 19144 dprdfeq0 19146 dprdlub 19150 |
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