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Mirrors > Home > HSE Home > Th. List > hfsmval | Structured version Visualization version GIF version |
Description: Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hfsmval | β’ ((π: ββΆβ β§ π: ββΆβ) β (π +fn π) = (π₯ β β β¦ ((πβπ₯) + (πβπ₯)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 11186 | . . 3 β’ β β V | |
2 | ax-hilex 30676 | . . 3 β’ β β V | |
3 | 1, 2 | elmap 8860 | . 2 β’ (π β (β βm β) β π: ββΆβ) |
4 | 1, 2 | elmap 8860 | . 2 β’ (π β (β βm β) β π: ββΆβ) |
5 | fveq1 6880 | . . . . 5 β’ (π = π β (πβπ₯) = (πβπ₯)) | |
6 | 5 | oveq1d 7416 | . . . 4 β’ (π = π β ((πβπ₯) + (πβπ₯)) = ((πβπ₯) + (πβπ₯))) |
7 | 6 | mpteq2dv 5240 | . . 3 β’ (π = π β (π₯ β β β¦ ((πβπ₯) + (πβπ₯))) = (π₯ β β β¦ ((πβπ₯) + (πβπ₯)))) |
8 | fveq1 6880 | . . . . 5 β’ (π = π β (πβπ₯) = (πβπ₯)) | |
9 | 8 | oveq2d 7417 | . . . 4 β’ (π = π β ((πβπ₯) + (πβπ₯)) = ((πβπ₯) + (πβπ₯))) |
10 | 9 | mpteq2dv 5240 | . . 3 β’ (π = π β (π₯ β β β¦ ((πβπ₯) + (πβπ₯))) = (π₯ β β β¦ ((πβπ₯) + (πβπ₯)))) |
11 | df-hfsum 31410 | . . 3 β’ +fn = (π β (β βm β), π β (β βm β) β¦ (π₯ β β β¦ ((πβπ₯) + (πβπ₯)))) | |
12 | 2 | mptex 7216 | . . 3 β’ (π₯ β β β¦ ((πβπ₯) + (πβπ₯))) β V |
13 | 7, 10, 11, 12 | ovmpo 7560 | . 2 β’ ((π β (β βm β) β§ π β (β βm β)) β (π +fn π) = (π₯ β β β¦ ((πβπ₯) + (πβπ₯)))) |
14 | 3, 4, 13 | syl2anbr 598 | 1 β’ ((π: ββΆβ β§ π: ββΆβ) β (π +fn π) = (π₯ β β β¦ ((πβπ₯) + (πβπ₯)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β¦ cmpt 5221 βΆwf 6529 βcfv 6533 (class class class)co 7401 βm cmap 8815 βcc 11103 + caddc 11108 βchba 30596 +fn chfs 30618 |
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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-hilex 30676 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-map 8817 df-hfsum 31410 |
This theorem is referenced by: hfsval 31420 |
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