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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 3464 | . . 3 β’ (πΌ β π β πΌ β V) | |
3 | oveq2 7366 | . . . . 5 β’ (π = πΌ β (βfld freeLMod π) = (βfld freeLMod πΌ)) | |
4 | 3 | fveq2d 6847 | . . . 4 β’ (π = πΌ β (toβPreHilβ(βfld freeLMod π)) = (toβPreHilβ(βfld freeLMod πΌ))) |
5 | df-rrx 24752 | . . . 4 β’ β^ = (π β V β¦ (toβPreHilβ(βfld freeLMod π))) | |
6 | fvex 6856 | . . . 4 β’ (toβPreHilβ(βfld freeLMod πΌ)) β V | |
7 | 4, 5, 6 | fvmpt 6949 | . . 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 2789 | 1 β’ (πΌ β π β π» = (toβPreHilβ(βfld freeLMod πΌ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3446 βcfv 6497 (class class class)co 7358 βfldcrefld 21011 freeLMod cfrlm 21155 toβPreHilctcph 24534 β^crrx 24750 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-rrx 24752 |
This theorem is referenced by: rrxbase 24755 rrxprds 24756 rrxnm 24758 rrxcph 24759 rrxds 24760 rrxvsca 24761 rrxplusgvscavalb 24762 rrx0 24764 rrxdim 32314 rrxtopn 44532 opnvonmbllem2 44881 |
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