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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 2622 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | tsmssub.1 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | tsmssub.2 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TopGrp) | |
5 | tgptmd 21883 | . . . 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 21882 | . . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) | |
10 | eqid 2622 | . . . . . 6 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
11 | 1, 10 | grpinvf 17466 | . . . . 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 6058 | . . . 4 ⊢ (((invg‘𝐺):𝐵⟶𝐵 ∧ 𝐻:𝐴⟶𝐵) → ((invg‘𝐺) ∘ 𝐻):𝐴⟶𝐵) | |
15 | 12, 13, 14 | syl2anc 693 | . . 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 21951 | . . 3 ⊢ (𝜑 → ((invg‘𝐺)‘𝑌) ∈ (𝐺 tsums ((invg‘𝐺) ∘ 𝐻))) |
19 | 1, 2, 3, 6, 7, 8, 15, 16, 18 | tsmsadd 21950 | . 2 ⊢ (𝜑 → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) ∈ (𝐺 tsums (𝐹 ∘𝑓 (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)))) |
20 | tgptps 21884 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp) | |
21 | 4, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TopSp) |
22 | 1, 3, 21, 7, 8 | tsmscl 21938 | . . . 4 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵) |
23 | 22, 16 | sseldd 3604 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
24 | 1, 3, 21, 7, 13 | tsmscl 21938 | . . . 4 ⊢ (𝜑 → (𝐺 tsums 𝐻) ⊆ 𝐵) |
25 | 24, 17 | sseldd 3604 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
26 | tsmssub.p | . . . 4 ⊢ − = (-g‘𝐺) | |
27 | 1, 2, 10, 26 | grpsubval 17465 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
28 | 23, 25, 27 | syl2anc 693 | . 2 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
29 | 8 | ffvelrnda 6359 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵) |
30 | 13 | ffvelrnda 6359 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐻‘𝑘) ∈ 𝐵) |
31 | 1, 2, 10, 26 | grpsubval 17465 | . . . . . 6 ⊢ (((𝐹‘𝑘) ∈ 𝐵 ∧ (𝐻‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘) − (𝐻‘𝑘)) = ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘)))) |
32 | 29, 30, 31 | syl2anc 693 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘) − (𝐻‘𝑘)) = ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘)))) |
33 | 32 | mpteq2dva 4744 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) − (𝐻‘𝑘))) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘))))) |
34 | 8 | feqmptd 6249 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
35 | 13 | feqmptd 6249 | . . . . 5 ⊢ (𝜑 → 𝐻 = (𝑘 ∈ 𝐴 ↦ (𝐻‘𝑘))) |
36 | 7, 29, 30, 34, 35 | offval2 6914 | . . . 4 ⊢ (𝜑 → (𝐹 ∘𝑓 − 𝐻) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) − (𝐻‘𝑘)))) |
37 | fvexd 6203 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((invg‘𝐺)‘(𝐻‘𝑘)) ∈ V) | |
38 | 12 | feqmptd 6249 | . . . . . 6 ⊢ (𝜑 → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ ((invg‘𝐺)‘𝑥))) |
39 | fveq2 6191 | . . . . . 6 ⊢ (𝑥 = (𝐻‘𝑘) → ((invg‘𝐺)‘𝑥) = ((invg‘𝐺)‘(𝐻‘𝑘))) | |
40 | 30, 35, 38, 39 | fmptco 6396 | . . . . 5 ⊢ (𝜑 → ((invg‘𝐺) ∘ 𝐻) = (𝑘 ∈ 𝐴 ↦ ((invg‘𝐺)‘(𝐻‘𝑘)))) |
41 | 7, 29, 37, 34, 40 | offval2 6914 | . . . 4 ⊢ (𝜑 → (𝐹 ∘𝑓 (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘))))) |
42 | 33, 36, 41 | 3eqtr4d 2666 | . . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 − 𝐻) = (𝐹 ∘𝑓 (+g‘𝐺)((invg‘𝐺) ∘ 𝐻))) |
43 | 42 | oveq2d 6666 | . 2 ⊢ (𝜑 → (𝐺 tsums (𝐹 ∘𝑓 − 𝐻)) = (𝐺 tsums (𝐹 ∘𝑓 (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)))) |
44 | 19, 28, 43 | 3eltr4d 2716 | 1 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝐺 tsums (𝐹 ∘𝑓 − 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ↦ cmpt 4729 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 Basecbs 15857 +gcplusg 15941 Grpcgrp 17422 invgcminusg 17423 -gcsg 17424 CMndccmn 18193 TopSpctps 20736 TopMndctmd 21874 TopGrpctgp 21875 tsums ctsu 21929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-gsum 16103 df-topgen 16104 df-plusf 17241 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-fbas 19743 df-fg 19744 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-ntr 20824 df-nei 20902 df-cn 21031 df-cnp 21032 df-tx 21365 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-tmd 21876 df-tgp 21877 df-tsms 21930 |
This theorem is referenced by: tgptsmscls 21953 |
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