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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 22969 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 2737 | . . . . . . . 8 β’ (BaseβπΊ) = (BaseβπΊ) | |
5 | 3, 4 | tmdtopon 23384 | . . . . . . 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 22552 | . . . . . 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 1371 | . . . . 5 β’ (π β (π₯ β π β¦ π΄):πβΆ(BaseβπΊ)) |
10 | 9 | fvmptelcdm 7057 | . . . 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 22552 | . . . . . 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 1371 | . . . . 5 β’ (π β (π₯ β π β¦ π΅):πβΆ(BaseβπΊ)) |
14 | 13 | fvmptelcdm 7057 | . . . 4 β’ ((π β§ π₯ β π) β π΅ β (BaseβπΊ)) |
15 | cnmpt1plusg.p | . . . . 5 β’ + = (+gβπΊ) | |
16 | eqid 2737 | . . . . 5 β’ (+πβπΊ) = (+πβπΊ) | |
17 | 4, 15, 16 | plusfval 18464 | . . . 4 β’ ((π΄ β (BaseβπΊ) β§ π΅ β (BaseβπΊ)) β (π΄(+πβπΊ)π΅) = (π΄ + π΅)) |
18 | 10, 14, 17 | syl2anc 584 | . . 3 β’ ((π β§ π₯ β π) β (π΄(+πβπΊ)π΅) = (π΄ + π΅)) |
19 | 18 | mpteq2dva 5203 | . 2 β’ (π β (π₯ β π β¦ (π΄(+πβπΊ)π΅)) = (π₯ β π β¦ (π΄ + π΅))) |
20 | 3, 16 | tmdcn 23386 | . . . 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 22969 | . 2 β’ (π β (π₯ β π β¦ (π΄(+πβπΊ)π΅)) β (πΎ Cn π½)) |
23 | 19, 22 | eqeltrrd 2839 | 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 396 = wceq 1541 β wcel 2106 β¦ cmpt 5186 βΆwf 6489 βcfv 6493 (class class class)co 7351 Basecbs 17043 +gcplusg 17093 TopOpenctopn 17263 +πcplusf 18454 TopOnctopon 22211 Cn ccn 22527 Γt ctx 22863 TopMndctmd 23373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-map 8725 df-topgen 17285 df-plusf 18456 df-top 22195 df-topon 22212 df-topsp 22234 df-bases 22248 df-cn 22530 df-tx 22865 df-tmd 23375 |
This theorem is referenced by: tmdmulg 23395 tmdgsum 23398 tmdlactcn 23405 clsnsg 23413 tgpt0 23422 cnmpt1mulr 23485 |
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