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Mirrors > Home > MPE Home > Th. List > cnmpt1plusg | Structured version Visualization version GIF version |
Description: Continuity of the group sum; analogue of cnmpt12f 23525 which cannot be used directly because +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
tgpcn.j | β’ π½ = (TopOpenβπΊ) |
cnmpt1plusg.p | β’ + = (+gβπΊ) |
cnmpt1plusg.g | β’ (π β πΊ β TopMnd) |
cnmpt1plusg.k | β’ (π β πΎ β (TopOnβπ)) |
cnmpt1plusg.a | β’ (π β (π₯ β π β¦ π΄) β (πΎ Cn π½)) |
cnmpt1plusg.b | β’ (π β (π₯ β π β¦ π΅) β (πΎ Cn π½)) |
Ref | Expression |
---|---|
cnmpt1plusg | β’ (π β (π₯ β π β¦ (π΄ + π΅)) β (πΎ Cn π½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmpt1plusg.k | . . . . . 6 β’ (π β πΎ β (TopOnβπ)) | |
2 | cnmpt1plusg.g | . . . . . . 7 β’ (π β πΊ β TopMnd) | |
3 | tgpcn.j | . . . . . . . 8 β’ π½ = (TopOpenβπΊ) | |
4 | eqid 2726 | . . . . . . . 8 β’ (BaseβπΊ) = (BaseβπΊ) | |
5 | 3, 4 | tmdtopon 23940 | . . . . . . 7 β’ (πΊ β TopMnd β π½ β (TopOnβ(BaseβπΊ))) |
6 | 2, 5 | syl 17 | . . . . . 6 β’ (π β π½ β (TopOnβ(BaseβπΊ))) |
7 | cnmpt1plusg.a | . . . . . 6 β’ (π β (π₯ β π β¦ π΄) β (πΎ Cn π½)) | |
8 | cnf2 23108 | . . . . . 6 β’ ((πΎ β (TopOnβπ) β§ π½ β (TopOnβ(BaseβπΊ)) β§ (π₯ β π β¦ π΄) β (πΎ Cn π½)) β (π₯ β π β¦ π΄):πβΆ(BaseβπΊ)) | |
9 | 1, 6, 7, 8 | syl3anc 1368 | . . . . 5 β’ (π β (π₯ β π β¦ π΄):πβΆ(BaseβπΊ)) |
10 | 9 | fvmptelcdm 7108 | . . . 4 β’ ((π β§ π₯ β π) β π΄ β (BaseβπΊ)) |
11 | cnmpt1plusg.b | . . . . . 6 β’ (π β (π₯ β π β¦ π΅) β (πΎ Cn π½)) | |
12 | cnf2 23108 | . . . . . 6 β’ ((πΎ β (TopOnβπ) β§ π½ β (TopOnβ(BaseβπΊ)) β§ (π₯ β π β¦ π΅) β (πΎ Cn π½)) β (π₯ β π β¦ π΅):πβΆ(BaseβπΊ)) | |
13 | 1, 6, 11, 12 | syl3anc 1368 | . . . . 5 β’ (π β (π₯ β π β¦ π΅):πβΆ(BaseβπΊ)) |
14 | 13 | fvmptelcdm 7108 | . . . 4 β’ ((π β§ π₯ β π) β π΅ β (BaseβπΊ)) |
15 | cnmpt1plusg.p | . . . . 5 β’ + = (+gβπΊ) | |
16 | eqid 2726 | . . . . 5 β’ (+πβπΊ) = (+πβπΊ) | |
17 | 4, 15, 16 | plusfval 18580 | . . . 4 β’ ((π΄ β (BaseβπΊ) β§ π΅ β (BaseβπΊ)) β (π΄(+πβπΊ)π΅) = (π΄ + π΅)) |
18 | 10, 14, 17 | syl2anc 583 | . . 3 β’ ((π β§ π₯ β π) β (π΄(+πβπΊ)π΅) = (π΄ + π΅)) |
19 | 18 | mpteq2dva 5241 | . 2 β’ (π β (π₯ β π β¦ (π΄(+πβπΊ)π΅)) = (π₯ β π β¦ (π΄ + π΅))) |
20 | 3, 16 | tmdcn 23942 | . . . 4 β’ (πΊ β TopMnd β (+πβπΊ) β ((π½ Γt π½) Cn π½)) |
21 | 2, 20 | syl 17 | . . 3 β’ (π β (+πβπΊ) β ((π½ Γt π½) Cn π½)) |
22 | 1, 7, 11, 21 | cnmpt12f 23525 | . 2 β’ (π β (π₯ β π β¦ (π΄(+πβπΊ)π΅)) β (πΎ Cn π½)) |
23 | 19, 22 | eqeltrrd 2828 | 1 β’ (π β (π₯ β π β¦ (π΄ + π΅)) β (πΎ Cn π½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β¦ cmpt 5224 βΆwf 6533 βcfv 6537 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 TopOpenctopn 17376 +πcplusf 18570 TopOnctopon 22767 Cn ccn 23083 Γt ctx 23419 TopMndctmd 23929 |
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-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-ral 3056 df-rex 3065 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-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 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-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-map 8824 df-topgen 17398 df-plusf 18572 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cn 23086 df-tx 23421 df-tmd 23931 |
This theorem is referenced by: tmdmulg 23951 tmdgsum 23954 tmdlactcn 23961 clsnsg 23969 tgpt0 23978 cnmpt1mulr 24041 |
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