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Mirrors > Home > HSE Home > Th. List > axhvdistr2-zf | Structured version Visualization version GIF version |
Description: Derive Axiom ax-hvdistr2 30000 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axhil.1 | β’ π = β¨β¨ +β , Β·β β©, normββ© |
axhil.2 | β’ π β CHilOLD |
Ref | Expression |
---|---|
axhvdistr2-zf | β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ + π΅) Β·β πΆ) = ((π΄ Β·β πΆ) +β (π΅ Β·β πΆ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axhil.2 | . 2 β’ π β CHilOLD | |
2 | df-hba 29960 | . . . 4 β’ β = (BaseSetββ¨β¨ +β , Β·β β©, normββ©) | |
3 | axhil.1 | . . . . 5 β’ π = β¨β¨ +β , Β·β β©, normββ© | |
4 | 3 | fveq2i 6849 | . . . 4 β’ (BaseSetβπ) = (BaseSetββ¨β¨ +β , Β·β β©, normββ©) |
5 | 2, 4 | eqtr4i 2764 | . . 3 β’ β = (BaseSetβπ) |
6 | 1 | hlnvi 29883 | . . . 4 β’ π β NrmCVec |
7 | 3, 6 | h2hva 29965 | . . 3 β’ +β = ( +π£ βπ) |
8 | 3, 6 | h2hsm 29966 | . . 3 β’ Β·β = ( Β·π OLD βπ) |
9 | 5, 7, 8 | hldir 29899 | . 2 β’ ((π β CHilOLD β§ (π΄ β β β§ π΅ β β β§ πΆ β β)) β ((π΄ + π΅) Β·β πΆ) = ((π΄ Β·β πΆ) +β (π΅ Β·β πΆ))) |
10 | 1, 9 | mpan 689 | 1 β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ + π΅) Β·β πΆ) = ((π΄ Β·β πΆ) +β (π΅ Β·β πΆ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 β¨cop 4596 βcfv 6500 (class class class)co 7361 βcc 11057 + caddc 11062 BaseSetcba 29577 CHilOLDchlo 29876 βchba 29910 +β cva 29911 Β·β csm 29912 normβcno 29914 |
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-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
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 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-1st 7925 df-2nd 7926 df-vc 29550 df-nv 29583 df-va 29586 df-ba 29587 df-sm 29588 df-0v 29589 df-nmcv 29591 df-cbn 29854 df-hlo 29877 df-hba 29960 |
This theorem is referenced by: (None) |
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