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Mirrors > Home > HSE Home > Th. List > lnfnsubi | Structured version Visualization version GIF version |
Description: Subtraction property for a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnfnl.1 | β’ π β LinFn |
Ref | Expression |
---|---|
lnfnsubi | β’ ((π΄ β β β§ π΅ β β) β (πβ(π΄ ββ π΅)) = ((πβπ΄) β (πβπ΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1cn 12188 | . . 3 β’ -1 β β | |
2 | lnfnl.1 | . . . 4 β’ π β LinFn | |
3 | 2 | lnfnaddmuli 30695 | . . 3 β’ ((-1 β β β§ π΄ β β β§ π΅ β β) β (πβ(π΄ +β (-1 Β·β π΅))) = ((πβπ΄) + (-1 Β· (πβπ΅)))) |
4 | 1, 3 | mp3an1 1447 | . 2 β’ ((π΄ β β β§ π΅ β β) β (πβ(π΄ +β (-1 Β·β π΅))) = ((πβπ΄) + (-1 Β· (πβπ΅)))) |
5 | hvsubval 29666 | . . 3 β’ ((π΄ β β β§ π΅ β β) β (π΄ ββ π΅) = (π΄ +β (-1 Β·β π΅))) | |
6 | 5 | fveq2d 6829 | . 2 β’ ((π΄ β β β§ π΅ β β) β (πβ(π΄ ββ π΅)) = (πβ(π΄ +β (-1 Β·β π΅)))) |
7 | 2 | lnfnfi 30691 | . . . 4 β’ π: ββΆβ |
8 | 7 | ffvelcdmi 7016 | . . 3 β’ (π΄ β β β (πβπ΄) β β) |
9 | 7 | ffvelcdmi 7016 | . . 3 β’ (π΅ β β β (πβπ΅) β β) |
10 | mulm1 11517 | . . . . . 6 β’ ((πβπ΅) β β β (-1 Β· (πβπ΅)) = -(πβπ΅)) | |
11 | 10 | oveq2d 7353 | . . . . 5 β’ ((πβπ΅) β β β ((πβπ΄) + (-1 Β· (πβπ΅))) = ((πβπ΄) + -(πβπ΅))) |
12 | 11 | adantl 482 | . . . 4 β’ (((πβπ΄) β β β§ (πβπ΅) β β) β ((πβπ΄) + (-1 Β· (πβπ΅))) = ((πβπ΄) + -(πβπ΅))) |
13 | negsub 11370 | . . . 4 β’ (((πβπ΄) β β β§ (πβπ΅) β β) β ((πβπ΄) + -(πβπ΅)) = ((πβπ΄) β (πβπ΅))) | |
14 | 12, 13 | eqtr2d 2777 | . . 3 β’ (((πβπ΄) β β β§ (πβπ΅) β β) β ((πβπ΄) β (πβπ΅)) = ((πβπ΄) + (-1 Β· (πβπ΅)))) |
15 | 8, 9, 14 | syl2an 596 | . 2 β’ ((π΄ β β β§ π΅ β β) β ((πβπ΄) β (πβπ΅)) = ((πβπ΄) + (-1 Β· (πβπ΅)))) |
16 | 4, 6, 15 | 3eqtr4d 2786 | 1 β’ ((π΄ β β β§ π΅ β β) β (πβ(π΄ ββ π΅)) = ((πβπ΄) β (πβπ΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1540 β wcel 2105 βcfv 6479 (class class class)co 7337 βcc 10970 1c1 10973 + caddc 10975 Β· cmul 10977 β cmin 11306 -cneg 11307 βchba 29569 +β cva 29570 Β·β csm 29571 ββ cmv 29575 LinFnclf 29604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-hilex 29649 ax-hv0cl 29653 ax-hvaddid 29654 ax-hfvmul 29655 ax-hvmulid 29656 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-po 5532 df-so 5533 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-ltxr 11115 df-sub 11308 df-neg 11309 df-hvsub 29621 df-lnfn 30498 |
This theorem is referenced by: lnfnconi 30705 riesz3i 30712 |
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