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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrhval | Structured version Visualization version GIF version |
Description: Value of the canonical homormorphism from the real numbers to a complete space. (Contributed by Thierry Arnoux, 2-Nov-2017.) |
Ref | Expression |
---|---|
rrhval.1 | โข ๐ฝ = (topGenโran (,)) |
rrhval.2 | โข ๐พ = (TopOpenโ๐ ) |
Ref | Expression |
---|---|
rrhval | โข (๐ โ ๐ โ (โHomโ๐ ) = ((๐ฝCnExt๐พ)โ(โHomโ๐ ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3493 | . 2 โข (๐ โ ๐ โ ๐ โ V) | |
2 | rrhval.1 | . . . . . . 7 โข ๐ฝ = (topGenโran (,)) | |
3 | 2 | eqcomi 2742 | . . . . . 6 โข (topGenโran (,)) = ๐ฝ |
4 | 3 | a1i 11 | . . . . 5 โข (๐ = ๐ โ (topGenโran (,)) = ๐ฝ) |
5 | fveq2 6892 | . . . . . 6 โข (๐ = ๐ โ (TopOpenโ๐) = (TopOpenโ๐ )) | |
6 | rrhval.2 | . . . . . 6 โข ๐พ = (TopOpenโ๐ ) | |
7 | 5, 6 | eqtr4di 2791 | . . . . 5 โข (๐ = ๐ โ (TopOpenโ๐) = ๐พ) |
8 | 4, 7 | oveq12d 7427 | . . . 4 โข (๐ = ๐ โ ((topGenโran (,))CnExt(TopOpenโ๐)) = (๐ฝCnExt๐พ)) |
9 | fveq2 6892 | . . . 4 โข (๐ = ๐ โ (โHomโ๐) = (โHomโ๐ )) | |
10 | 8, 9 | fveq12d 6899 | . . 3 โข (๐ = ๐ โ (((topGenโran (,))CnExt(TopOpenโ๐))โ(โHomโ๐)) = ((๐ฝCnExt๐พ)โ(โHomโ๐ ))) |
11 | df-rrh 32975 | . . 3 โข โHom = (๐ โ V โฆ (((topGenโran (,))CnExt(TopOpenโ๐))โ(โHomโ๐))) | |
12 | fvex 6905 | . . 3 โข ((๐ฝCnExt๐พ)โ(โHomโ๐ )) โ V | |
13 | 10, 11, 12 | fvmpt 6999 | . 2 โข (๐ โ V โ (โHomโ๐ ) = ((๐ฝCnExt๐พ)โ(โHomโ๐ ))) |
14 | 1, 13 | syl 17 | 1 โข (๐ โ ๐ โ (โHomโ๐ ) = ((๐ฝCnExt๐พ)โ(โHomโ๐ ))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 Vcvv 3475 ran crn 5678 โcfv 6544 (class class class)co 7409 (,)cioo 13324 TopOpenctopn 17367 topGenctg 17383 CnExtccnext 23563 โHomcqqh 32952 โHomcrrh 32973 |
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-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-rrh 32975 |
This theorem is referenced by: rrhcn 32977 rrhqima 32994 rrhre 33001 |
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