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Mirrors > Home > MPE Home > Th. List > tsmsi | Structured version Visualization version GIF version |
Description: The property of being a sum of the sequence πΉ in the topological commutative monoid πΊ. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
eltsms.b | β’ π΅ = (BaseβπΊ) |
eltsms.j | β’ π½ = (TopOpenβπΊ) |
eltsms.s | β’ π = (π« π΄ β© Fin) |
eltsms.1 | β’ (π β πΊ β CMnd) |
eltsms.2 | β’ (π β πΊ β TopSp) |
eltsms.a | β’ (π β π΄ β π) |
eltsms.f | β’ (π β πΉ:π΄βΆπ΅) |
tsmsi.3 | β’ (π β πΆ β (πΊ tsums πΉ)) |
tsmsi.4 | β’ (π β π β π½) |
tsmsi.5 | β’ (π β πΆ β π) |
Ref | Expression |
---|---|
tsmsi | β’ (π β βπ§ β π βπ¦ β π (π§ β π¦ β (πΊ Ξ£g (πΉ βΎ π¦)) β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsmsi.5 | . 2 β’ (π β πΆ β π) | |
2 | eleq2 2823 | . . . 4 β’ (π’ = π β (πΆ β π’ β πΆ β π)) | |
3 | eleq2 2823 | . . . . . 6 β’ (π’ = π β ((πΊ Ξ£g (πΉ βΎ π¦)) β π’ β (πΊ Ξ£g (πΉ βΎ π¦)) β π)) | |
4 | 3 | imbi2d 341 | . . . . 5 β’ (π’ = π β ((π§ β π¦ β (πΊ Ξ£g (πΉ βΎ π¦)) β π’) β (π§ β π¦ β (πΊ Ξ£g (πΉ βΎ π¦)) β π))) |
5 | 4 | rexralbidv 3211 | . . . 4 β’ (π’ = π β (βπ§ β π βπ¦ β π (π§ β π¦ β (πΊ Ξ£g (πΉ βΎ π¦)) β π’) β βπ§ β π βπ¦ β π (π§ β π¦ β (πΊ Ξ£g (πΉ βΎ π¦)) β π))) |
6 | 2, 5 | imbi12d 345 | . . 3 β’ (π’ = π β ((πΆ β π’ β βπ§ β π βπ¦ β π (π§ β π¦ β (πΊ Ξ£g (πΉ βΎ π¦)) β π’)) β (πΆ β π β βπ§ β π βπ¦ β π (π§ β π¦ β (πΊ Ξ£g (πΉ βΎ π¦)) β π)))) |
7 | tsmsi.3 | . . . . 5 β’ (π β πΆ β (πΊ tsums πΉ)) | |
8 | eltsms.b | . . . . . 6 β’ π΅ = (BaseβπΊ) | |
9 | eltsms.j | . . . . . 6 β’ π½ = (TopOpenβπΊ) | |
10 | eltsms.s | . . . . . 6 β’ π = (π« π΄ β© Fin) | |
11 | eltsms.1 | . . . . . 6 β’ (π β πΊ β CMnd) | |
12 | eltsms.2 | . . . . . 6 β’ (π β πΊ β TopSp) | |
13 | eltsms.a | . . . . . 6 β’ (π β π΄ β π) | |
14 | eltsms.f | . . . . . 6 β’ (π β πΉ:π΄βΆπ΅) | |
15 | 8, 9, 10, 11, 12, 13, 14 | eltsms 23500 | . . . . 5 β’ (π β (πΆ β (πΊ tsums πΉ) β (πΆ β π΅ β§ βπ’ β π½ (πΆ β π’ β βπ§ β π βπ¦ β π (π§ β π¦ β (πΊ Ξ£g (πΉ βΎ π¦)) β π’))))) |
16 | 7, 15 | mpbid 231 | . . . 4 β’ (π β (πΆ β π΅ β§ βπ’ β π½ (πΆ β π’ β βπ§ β π βπ¦ β π (π§ β π¦ β (πΊ Ξ£g (πΉ βΎ π¦)) β π’)))) |
17 | 16 | simprd 497 | . . 3 β’ (π β βπ’ β π½ (πΆ β π’ β βπ§ β π βπ¦ β π (π§ β π¦ β (πΊ Ξ£g (πΉ βΎ π¦)) β π’))) |
18 | tsmsi.4 | . . 3 β’ (π β π β π½) | |
19 | 6, 17, 18 | rspcdva 3581 | . 2 β’ (π β (πΆ β π β βπ§ β π βπ¦ β π (π§ β π¦ β (πΊ Ξ£g (πΉ βΎ π¦)) β π))) |
20 | 1, 19 | mpd 15 | 1 β’ (π β βπ§ β π βπ¦ β π (π§ β π¦ β (πΊ Ξ£g (πΉ βΎ π¦)) β π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 βwrex 3070 β© cin 3910 β wss 3911 π« cpw 4561 βΎ cres 5636 βΆwf 6493 βcfv 6497 (class class class)co 7358 Fincfn 8886 Basecbs 17088 TopOpenctopn 17308 Ξ£g cgsu 17327 CMndccmn 19567 TopSpctps 22297 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-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-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 df-seq 13913 df-hash 14237 df-0g 17328 df-gsum 17329 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-cntz 19102 df-cmn 19569 df-fbas 20809 df-fg 20810 df-top 22259 df-topon 22276 df-topsp 22298 df-ntr 22387 df-nei 22465 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-tsms 23494 |
This theorem is referenced by: tsmsxplem1 23520 |
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