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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 23586 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 2725 | . . . . . . . 8 β’ (BaseβπΊ) = (BaseβπΊ) | |
5 | 3, 4 | tmdtopon 24001 | . . . . . . 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 23169 | . . . . . 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 7117 | . . . 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 23169 | . . . . . 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 7117 | . . . 4 β’ ((π β§ π₯ β π) β π΅ β (BaseβπΊ)) |
15 | cnmpt1plusg.p | . . . . 5 β’ + = (+gβπΊ) | |
16 | eqid 2725 | . . . . 5 β’ (+πβπΊ) = (+πβπΊ) | |
17 | 4, 15, 16 | plusfval 18604 | . . . 4 β’ ((π΄ β (BaseβπΊ) β§ π΅ β (BaseβπΊ)) β (π΄(+πβπΊ)π΅) = (π΄ + π΅)) |
18 | 10, 14, 17 | syl2anc 582 | . . 3 β’ ((π β§ π₯ β π) β (π΄(+πβπΊ)π΅) = (π΄ + π΅)) |
19 | 18 | mpteq2dva 5243 | . 2 β’ (π β (π₯ β π β¦ (π΄(+πβπΊ)π΅)) = (π₯ β π β¦ (π΄ + π΅))) |
20 | 3, 16 | tmdcn 24003 | . . . 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 23586 | . 2 β’ (π β (π₯ β π β¦ (π΄(+πβπΊ)π΅)) β (πΎ Cn π½)) |
23 | 19, 22 | eqeltrrd 2826 | 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 394 = wceq 1533 β wcel 2098 β¦ cmpt 5226 βΆwf 6538 βcfv 6542 (class class class)co 7415 Basecbs 17177 +gcplusg 17230 TopOpenctopn 17400 +πcplusf 18594 TopOnctopon 22828 Cn ccn 23144 Γt ctx 23480 TopMndctmd 23990 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1st 7989 df-2nd 7990 df-map 8843 df-topgen 17422 df-plusf 18596 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22865 df-cn 23147 df-tx 23482 df-tmd 23992 |
This theorem is referenced by: tmdmulg 24012 tmdgsum 24015 tmdlactcn 24022 clsnsg 24030 tgpt0 24039 cnmpt1mulr 24102 |
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