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Mirrors > Home > HSE Home > Th. List > axhv0cl-zf | Structured version Visualization version GIF version |
Description: Derive Axiom ax-hv0cl 30251 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 |
---|---|
axhv0cl-zf | β’ 0β β β |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axhil.2 | . 2 β’ π β CHilOLD | |
2 | df-hba 30217 | . . . 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 6894 | . . . 4 β’ (BaseSetβπ) = (BaseSetββ¨β¨ +β , Β·β β©, normββ©) |
5 | 2, 4 | eqtr4i 2763 | . . 3 β’ β = (BaseSetβπ) |
6 | df-h0v 30218 | . . . 4 β’ 0β = (0vecββ¨β¨ +β , Β·β β©, normββ©) | |
7 | 3 | fveq2i 6894 | . . . 4 β’ (0vecβπ) = (0vecββ¨β¨ +β , Β·β β©, normββ©) |
8 | 6, 7 | eqtr4i 2763 | . . 3 β’ 0β = (0vecβπ) |
9 | 5, 8 | hl0cl 30150 | . 2 β’ (π β CHilOLD β 0β β β) |
10 | 1, 9 | ax-mp 5 | 1 β’ 0β β β |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 β wcel 2106 β¨cop 4634 βcfv 6543 BaseSetcba 29834 0veccn0v 29836 CHilOLDchlo 30133 βchba 30167 +β cva 30168 Β·β csm 30169 normβcno 30171 0βc0v 30172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-1st 7974 df-2nd 7975 df-grpo 29741 df-gid 29742 df-ablo 29793 df-vc 29807 df-nv 29840 df-va 29843 df-ba 29844 df-sm 29845 df-0v 29846 df-nmcv 29848 df-cbn 30111 df-hlo 30134 df-hba 30217 df-h0v 30218 |
This theorem is referenced by: (None) |
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