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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 2735 | . . 3 ⊢ (Base‘(𝐺 ↾s 𝑆)) = (Base‘(𝐺 ↾s 𝑆)) | |
2 | eqid 2735 | . . 3 ⊢ (0g‘(𝐺 ↾s 𝑆)) = (0g‘(𝐺 ↾s 𝑆)) | |
3 | eqid 2735 | . . 3 ⊢ (Cntz‘(𝐺 ↾s 𝑆)) = (Cntz‘(𝐺 ↾s 𝑆)) | |
4 | gsumzsubmcl.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) | |
5 | eqid 2735 | . . . . 5 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆) | |
6 | 5 | submmnd 18839 | . . . 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 18840 | . . . . . 6 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
11 | 4, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
12 | 11 | feq3d 6724 | . . . 4 ⊢ (𝜑 → (𝐹:𝐴⟶𝑆 ↔ 𝐹:𝐴⟶(Base‘(𝐺 ↾s 𝑆)))) |
13 | 9, 12 | mpbid 232 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘(𝐺 ↾s 𝑆))) |
14 | gsumzsubmcl.c | . . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) | |
15 | 9 | frnd 6745 | . . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑆) |
16 | 14, 15 | ssind 4249 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ((𝑍‘ran 𝐹) ∩ 𝑆)) |
17 | gsumzsubmcl.z | . . . . . 6 ⊢ 𝑍 = (Cntz‘𝐺) | |
18 | 5, 17, 3 | resscntz 19364 | . . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ ran 𝐹 ⊆ 𝑆) → ((Cntz‘(𝐺 ↾s 𝑆))‘ran 𝐹) = ((𝑍‘ran 𝐹) ∩ 𝑆)) |
19 | 4, 15, 18 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((Cntz‘(𝐺 ↾s 𝑆))‘ran 𝐹) = ((𝑍‘ran 𝐹) ∩ 𝑆)) |
20 | 16, 19 | sseqtrrd 4037 | . . 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 18841 | . . . . 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 5174 | . . 3 ⊢ (𝜑 → 𝐹 finSupp (0g‘(𝐺 ↾s 𝑆))) |
26 | 1, 2, 3, 7, 8, 13, 20, 25 | gsumzcl 19944 | . 2 ⊢ (𝜑 → ((𝐺 ↾s 𝑆) Σg 𝐹) ∈ (Base‘(𝐺 ↾s 𝑆))) |
27 | 8, 4, 9, 5 | gsumsubm 18861 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 ↾s 𝑆) Σg 𝐹)) |
28 | 26, 27, 11 | 3eltr4d 2854 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 class class class wbr 5148 ran crn 5690 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 finSupp cfsupp 9399 Basecbs 17245 ↾s cress 17274 0gc0g 17486 Σg cgsu 17487 Mndcmnd 18760 SubMndcsubmnd 18808 Cntzccntz 19346 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-gsum 17489 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-cntz 19348 |
This theorem is referenced by: gsumsubmcl 19952 gsumzadd 19955 dprdfadd 20055 dprdfeq0 20057 dprdlub 20061 |
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