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Mirrors > Home > MPE Home > Th. List > tsmssub | Structured version Visualization version GIF version |
Description: The difference of two infinite group sums. (Contributed by Mario Carneiro, 20-Sep-2015.) |
Ref | Expression |
---|---|
tsmssub.b | ⊢ 𝐵 = (Base‘𝐺) |
tsmssub.p | ⊢ − = (-g‘𝐺) |
tsmssub.1 | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
tsmssub.2 | ⊢ (𝜑 → 𝐺 ∈ TopGrp) |
tsmssub.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
tsmssub.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
tsmssub.h | ⊢ (𝜑 → 𝐻:𝐴⟶𝐵) |
tsmssub.x | ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹)) |
tsmssub.y | ⊢ (𝜑 → 𝑌 ∈ (𝐺 tsums 𝐻)) |
Ref | Expression |
---|---|
tsmssub | ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝐺 tsums (𝐹 ∘f − 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsmssub.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2728 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | tsmssub.1 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | tsmssub.2 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TopGrp) | |
5 | tgptmd 24003 | . . . 4 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ TopMnd) |
7 | tsmssub.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | tsmssub.f | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
9 | tgpgrp 24002 | . . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) | |
10 | eqid 2728 | . . . . . 6 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
11 | 1, 10 | grpinvf 18950 | . . . . 5 ⊢ (𝐺 ∈ Grp → (invg‘𝐺):𝐵⟶𝐵) |
12 | 4, 9, 11 | 3syl 18 | . . . 4 ⊢ (𝜑 → (invg‘𝐺):𝐵⟶𝐵) |
13 | tsmssub.h | . . . 4 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵) | |
14 | fco 6752 | . . . 4 ⊢ (((invg‘𝐺):𝐵⟶𝐵 ∧ 𝐻:𝐴⟶𝐵) → ((invg‘𝐺) ∘ 𝐻):𝐴⟶𝐵) | |
15 | 12, 13, 14 | syl2anc 582 | . . 3 ⊢ (𝜑 → ((invg‘𝐺) ∘ 𝐻):𝐴⟶𝐵) |
16 | tsmssub.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹)) | |
17 | tsmssub.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝐺 tsums 𝐻)) | |
18 | 1, 10, 3, 4, 7, 13, 17 | tsmsinv 24072 | . . 3 ⊢ (𝜑 → ((invg‘𝐺)‘𝑌) ∈ (𝐺 tsums ((invg‘𝐺) ∘ 𝐻))) |
19 | 1, 2, 3, 6, 7, 8, 15, 16, 18 | tsmsadd 24071 | . 2 ⊢ (𝜑 → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) ∈ (𝐺 tsums (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)))) |
20 | tgptps 24004 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp) | |
21 | 4, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TopSp) |
22 | 1, 3, 21, 7, 8 | tsmscl 24059 | . . . 4 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵) |
23 | 22, 16 | sseldd 3983 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
24 | 1, 3, 21, 7, 13 | tsmscl 24059 | . . . 4 ⊢ (𝜑 → (𝐺 tsums 𝐻) ⊆ 𝐵) |
25 | 24, 17 | sseldd 3983 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
26 | tsmssub.p | . . . 4 ⊢ − = (-g‘𝐺) | |
27 | 1, 2, 10, 26 | grpsubval 18949 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
28 | 23, 25, 27 | syl2anc 582 | . 2 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
29 | 8 | ffvelcdmda 7099 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵) |
30 | 13 | ffvelcdmda 7099 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐻‘𝑘) ∈ 𝐵) |
31 | 1, 2, 10, 26 | grpsubval 18949 | . . . . . 6 ⊢ (((𝐹‘𝑘) ∈ 𝐵 ∧ (𝐻‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘) − (𝐻‘𝑘)) = ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘)))) |
32 | 29, 30, 31 | syl2anc 582 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘) − (𝐻‘𝑘)) = ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘)))) |
33 | 32 | mpteq2dva 5252 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) − (𝐻‘𝑘))) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘))))) |
34 | 8 | feqmptd 6972 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
35 | 13 | feqmptd 6972 | . . . . 5 ⊢ (𝜑 → 𝐻 = (𝑘 ∈ 𝐴 ↦ (𝐻‘𝑘))) |
36 | 7, 29, 30, 34, 35 | offval2 7711 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f − 𝐻) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) − (𝐻‘𝑘)))) |
37 | fvexd 6917 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((invg‘𝐺)‘(𝐻‘𝑘)) ∈ V) | |
38 | 12 | feqmptd 6972 | . . . . . 6 ⊢ (𝜑 → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ ((invg‘𝐺)‘𝑥))) |
39 | fveq2 6902 | . . . . . 6 ⊢ (𝑥 = (𝐻‘𝑘) → ((invg‘𝐺)‘𝑥) = ((invg‘𝐺)‘(𝐻‘𝑘))) | |
40 | 30, 35, 38, 39 | fmptco 7144 | . . . . 5 ⊢ (𝜑 → ((invg‘𝐺) ∘ 𝐻) = (𝑘 ∈ 𝐴 ↦ ((invg‘𝐺)‘(𝐻‘𝑘)))) |
41 | 7, 29, 37, 34, 40 | offval2 7711 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘))))) |
42 | 33, 36, 41 | 3eqtr4d 2778 | . . 3 ⊢ (𝜑 → (𝐹 ∘f − 𝐻) = (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻))) |
43 | 42 | oveq2d 7442 | . 2 ⊢ (𝜑 → (𝐺 tsums (𝐹 ∘f − 𝐻)) = (𝐺 tsums (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)))) |
44 | 19, 28, 43 | 3eltr4d 2844 | 1 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝐺 tsums (𝐹 ∘f − 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ↦ cmpt 5235 ∘ ccom 5686 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ∘f cof 7689 Basecbs 17187 +gcplusg 17240 Grpcgrp 18897 invgcminusg 18898 -gcsg 18899 CMndccmn 19742 TopSpctps 22854 TopMndctmd 23994 TopGrpctgp 23995 tsums ctsu 24050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-0g 17430 df-gsum 17431 df-topgen 17432 df-plusf 18606 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-ghm 19175 df-cntz 19275 df-cmn 19744 df-abl 19745 df-fbas 21283 df-fg 21284 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-ntr 22944 df-nei 23022 df-cn 23151 df-cnp 23152 df-tx 23486 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-tmd 23996 df-tgp 23997 df-tsms 24051 |
This theorem is referenced by: tgptsmscls 24074 |
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