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Mirrors > Home > MPE Home > Th. List > rrxval | Structured version Visualization version GIF version |
Description: Value of the generalized Euclidean space. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
Ref | Expression |
---|---|
rrxval.r | β’ π» = (β^βπΌ) |
Ref | Expression |
---|---|
rrxval | β’ (πΌ β π β π» = (toβPreHilβ(βfld freeLMod πΌ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxval.r | . 2 β’ π» = (β^βπΌ) | |
2 | elex 3492 | . . 3 β’ (πΌ β π β πΌ β V) | |
3 | oveq2 7434 | . . . . 5 β’ (π = πΌ β (βfld freeLMod π) = (βfld freeLMod πΌ)) | |
4 | 3 | fveq2d 6906 | . . . 4 β’ (π = πΌ β (toβPreHilβ(βfld freeLMod π)) = (toβPreHilβ(βfld freeLMod πΌ))) |
5 | df-rrx 25333 | . . . 4 β’ β^ = (π β V β¦ (toβPreHilβ(βfld freeLMod π))) | |
6 | fvex 6915 | . . . 4 β’ (toβPreHilβ(βfld freeLMod πΌ)) β V | |
7 | 4, 5, 6 | fvmpt 7010 | . . 3 β’ (πΌ β V β (β^βπΌ) = (toβPreHilβ(βfld freeLMod πΌ))) |
8 | 2, 7 | syl 17 | . 2 β’ (πΌ β π β (β^βπΌ) = (toβPreHilβ(βfld freeLMod πΌ))) |
9 | 1, 8 | eqtrid 2780 | 1 β’ (πΌ β π β π» = (toβPreHilβ(βfld freeLMod πΌ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3473 βcfv 6553 (class class class)co 7426 βfldcrefld 21543 freeLMod cfrlm 21687 toβPreHilctcph 25115 β^crrx 25331 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7429 df-rrx 25333 |
This theorem is referenced by: rrxbase 25336 rrxprds 25337 rrxnm 25339 rrxcph 25340 rrxds 25341 rrxvsca 25342 rrxplusgvscavalb 25343 rrx0 25345 rrxdim 33345 rrxtopn 45701 opnvonmbllem2 46050 |
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