![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2733 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | tsmssub.1 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | tsmssub.2 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TopGrp) | |
5 | tgptmd 23446 | . . . 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 23445 | . . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) | |
10 | eqid 2733 | . . . . . 6 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
11 | 1, 10 | grpinvf 18802 | . . . . 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 6693 | . . . 4 ⊢ (((invg‘𝐺):𝐵⟶𝐵 ∧ 𝐻:𝐴⟶𝐵) → ((invg‘𝐺) ∘ 𝐻):𝐴⟶𝐵) | |
15 | 12, 13, 14 | syl2anc 585 | . . 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 23515 | . . 3 ⊢ (𝜑 → ((invg‘𝐺)‘𝑌) ∈ (𝐺 tsums ((invg‘𝐺) ∘ 𝐻))) |
19 | 1, 2, 3, 6, 7, 8, 15, 16, 18 | tsmsadd 23514 | . 2 ⊢ (𝜑 → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) ∈ (𝐺 tsums (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)))) |
20 | tgptps 23447 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp) | |
21 | 4, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TopSp) |
22 | 1, 3, 21, 7, 8 | tsmscl 23502 | . . . 4 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵) |
23 | 22, 16 | sseldd 3946 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
24 | 1, 3, 21, 7, 13 | tsmscl 23502 | . . . 4 ⊢ (𝜑 → (𝐺 tsums 𝐻) ⊆ 𝐵) |
25 | 24, 17 | sseldd 3946 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
26 | tsmssub.p | . . . 4 ⊢ − = (-g‘𝐺) | |
27 | 1, 2, 10, 26 | grpsubval 18801 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
28 | 23, 25, 27 | syl2anc 585 | . 2 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
29 | 8 | ffvelcdmda 7036 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵) |
30 | 13 | ffvelcdmda 7036 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐻‘𝑘) ∈ 𝐵) |
31 | 1, 2, 10, 26 | grpsubval 18801 | . . . . . 6 ⊢ (((𝐹‘𝑘) ∈ 𝐵 ∧ (𝐻‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘) − (𝐻‘𝑘)) = ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘)))) |
32 | 29, 30, 31 | syl2anc 585 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘) − (𝐻‘𝑘)) = ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘)))) |
33 | 32 | mpteq2dva 5206 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) − (𝐻‘𝑘))) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘))))) |
34 | 8 | feqmptd 6911 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
35 | 13 | feqmptd 6911 | . . . . 5 ⊢ (𝜑 → 𝐻 = (𝑘 ∈ 𝐴 ↦ (𝐻‘𝑘))) |
36 | 7, 29, 30, 34, 35 | offval2 7638 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f − 𝐻) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) − (𝐻‘𝑘)))) |
37 | fvexd 6858 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((invg‘𝐺)‘(𝐻‘𝑘)) ∈ V) | |
38 | 12 | feqmptd 6911 | . . . . . 6 ⊢ (𝜑 → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ ((invg‘𝐺)‘𝑥))) |
39 | fveq2 6843 | . . . . . 6 ⊢ (𝑥 = (𝐻‘𝑘) → ((invg‘𝐺)‘𝑥) = ((invg‘𝐺)‘(𝐻‘𝑘))) | |
40 | 30, 35, 38, 39 | fmptco 7076 | . . . . 5 ⊢ (𝜑 → ((invg‘𝐺) ∘ 𝐻) = (𝑘 ∈ 𝐴 ↦ ((invg‘𝐺)‘(𝐻‘𝑘)))) |
41 | 7, 29, 37, 34, 40 | offval2 7638 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘))))) |
42 | 33, 36, 41 | 3eqtr4d 2783 | . . 3 ⊢ (𝜑 → (𝐹 ∘f − 𝐻) = (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻))) |
43 | 42 | oveq2d 7374 | . 2 ⊢ (𝜑 → (𝐺 tsums (𝐹 ∘f − 𝐻)) = (𝐺 tsums (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)))) |
44 | 19, 28, 43 | 3eltr4d 2849 | 1 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝐺 tsums (𝐹 ∘f − 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ↦ cmpt 5189 ∘ ccom 5638 ⟶wf 6493 ‘cfv 6497 (class class class)co 7358 ∘f cof 7616 Basecbs 17088 +gcplusg 17138 Grpcgrp 18753 invgcminusg 18754 -gcsg 18755 CMndccmn 19567 TopSpctps 22297 TopMndctmd 23437 TopGrpctgp 23438 tsums ctsu 23493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 df-seq 13913 df-hash 14237 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-0g 17328 df-gsum 17329 df-topgen 17330 df-plusf 18501 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-submnd 18607 df-grp 18756 df-minusg 18757 df-sbg 18758 df-ghm 19011 df-cntz 19102 df-cmn 19569 df-abl 19570 df-fbas 20809 df-fg 20810 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-ntr 22387 df-nei 22465 df-cn 22594 df-cnp 22595 df-tx 22929 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-tmd 23439 df-tgp 23440 df-tsms 23494 |
This theorem is referenced by: tgptsmscls 23517 |
Copyright terms: Public domain | W3C validator |