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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 2726 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | tsmssub.1 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | tsmssub.2 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TopGrp) | |
5 | tgptmd 23933 | . . . 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 23932 | . . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) | |
10 | eqid 2726 | . . . . . 6 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
11 | 1, 10 | grpinvf 18913 | . . . . 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 6734 | . . . 4 ⊢ (((invg‘𝐺):𝐵⟶𝐵 ∧ 𝐻:𝐴⟶𝐵) → ((invg‘𝐺) ∘ 𝐻):𝐴⟶𝐵) | |
15 | 12, 13, 14 | syl2anc 583 | . . 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 24002 | . . 3 ⊢ (𝜑 → ((invg‘𝐺)‘𝑌) ∈ (𝐺 tsums ((invg‘𝐺) ∘ 𝐻))) |
19 | 1, 2, 3, 6, 7, 8, 15, 16, 18 | tsmsadd 24001 | . 2 ⊢ (𝜑 → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) ∈ (𝐺 tsums (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)))) |
20 | tgptps 23934 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp) | |
21 | 4, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TopSp) |
22 | 1, 3, 21, 7, 8 | tsmscl 23989 | . . . 4 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵) |
23 | 22, 16 | sseldd 3978 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
24 | 1, 3, 21, 7, 13 | tsmscl 23989 | . . . 4 ⊢ (𝜑 → (𝐺 tsums 𝐻) ⊆ 𝐵) |
25 | 24, 17 | sseldd 3978 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
26 | tsmssub.p | . . . 4 ⊢ − = (-g‘𝐺) | |
27 | 1, 2, 10, 26 | grpsubval 18912 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
28 | 23, 25, 27 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
29 | 8 | ffvelcdmda 7079 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵) |
30 | 13 | ffvelcdmda 7079 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐻‘𝑘) ∈ 𝐵) |
31 | 1, 2, 10, 26 | grpsubval 18912 | . . . . . 6 ⊢ (((𝐹‘𝑘) ∈ 𝐵 ∧ (𝐻‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘) − (𝐻‘𝑘)) = ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘)))) |
32 | 29, 30, 31 | syl2anc 583 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘) − (𝐻‘𝑘)) = ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘)))) |
33 | 32 | mpteq2dva 5241 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) − (𝐻‘𝑘))) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘))))) |
34 | 8 | feqmptd 6953 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
35 | 13 | feqmptd 6953 | . . . . 5 ⊢ (𝜑 → 𝐻 = (𝑘 ∈ 𝐴 ↦ (𝐻‘𝑘))) |
36 | 7, 29, 30, 34, 35 | offval2 7686 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f − 𝐻) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) − (𝐻‘𝑘)))) |
37 | fvexd 6899 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((invg‘𝐺)‘(𝐻‘𝑘)) ∈ V) | |
38 | 12 | feqmptd 6953 | . . . . . 6 ⊢ (𝜑 → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ ((invg‘𝐺)‘𝑥))) |
39 | fveq2 6884 | . . . . . 6 ⊢ (𝑥 = (𝐻‘𝑘) → ((invg‘𝐺)‘𝑥) = ((invg‘𝐺)‘(𝐻‘𝑘))) | |
40 | 30, 35, 38, 39 | fmptco 7122 | . . . . 5 ⊢ (𝜑 → ((invg‘𝐺) ∘ 𝐻) = (𝑘 ∈ 𝐴 ↦ ((invg‘𝐺)‘(𝐻‘𝑘)))) |
41 | 7, 29, 37, 34, 40 | offval2 7686 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘))))) |
42 | 33, 36, 41 | 3eqtr4d 2776 | . . 3 ⊢ (𝜑 → (𝐹 ∘f − 𝐻) = (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻))) |
43 | 42 | oveq2d 7420 | . 2 ⊢ (𝜑 → (𝐺 tsums (𝐹 ∘f − 𝐻)) = (𝐺 tsums (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)))) |
44 | 19, 28, 43 | 3eltr4d 2842 | 1 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝐺 tsums (𝐹 ∘f − 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ↦ cmpt 5224 ∘ ccom 5673 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 ∘f cof 7664 Basecbs 17150 +gcplusg 17203 Grpcgrp 18860 invgcminusg 18861 -gcsg 18862 CMndccmn 19697 TopSpctps 22784 TopMndctmd 23924 TopGrpctgp 23925 tsums ctsu 23980 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-seq 13970 df-hash 14293 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-0g 17393 df-gsum 17394 df-topgen 17395 df-plusf 18569 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18710 df-submnd 18711 df-grp 18863 df-minusg 18864 df-sbg 18865 df-ghm 19136 df-cntz 19230 df-cmn 19699 df-abl 19700 df-fbas 21232 df-fg 21233 df-top 22746 df-topon 22763 df-topsp 22785 df-bases 22799 df-ntr 22874 df-nei 22952 df-cn 23081 df-cnp 23082 df-tx 23416 df-fil 23700 df-fm 23792 df-flim 23793 df-flf 23794 df-tmd 23926 df-tgp 23927 df-tsms 23981 |
This theorem is referenced by: tgptsmscls 24004 |
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