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Mirrors > Home > HSE Home > Th. List > hh0v | Structured version Visualization version GIF version |
Description: The zero vector of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhnv.1 | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
Ref | Expression |
---|---|
hh0v | ⊢ 0ℎ = (0vec‘𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hhnv.1 | . . . 4 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
2 | 1 | hhnv 28621 | . . 3 ⊢ 𝑈 ∈ NrmCVec |
3 | eqid 2793 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
4 | eqid 2793 | . . . 4 ⊢ (0vec‘𝑈) = (0vec‘𝑈) | |
5 | 3, 4 | 0vfval 28062 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (0vec‘𝑈) = (GId‘( +𝑣 ‘𝑈))) |
6 | 2, 5 | ax-mp 5 | . 2 ⊢ (0vec‘𝑈) = (GId‘( +𝑣 ‘𝑈)) |
7 | 1 | hhva 28622 | . . 3 ⊢ +ℎ = ( +𝑣 ‘𝑈) |
8 | 7 | fveq2i 6533 | . 2 ⊢ (GId‘ +ℎ ) = (GId‘( +𝑣 ‘𝑈)) |
9 | hilid 28617 | . 2 ⊢ (GId‘ +ℎ ) = 0ℎ | |
10 | 6, 8, 9 | 3eqtr2ri 2824 | 1 ⊢ 0ℎ = (0vec‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1520 ∈ wcel 2079 〈cop 4472 ‘cfv 6217 GIdcgi 27946 NrmCVeccnv 28040 +𝑣 cpv 28041 0veccn0v 28044 +ℎ cva 28376 ·ℎ csm 28377 normℎcno 28379 0ℎc0v 28380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 ax-hilex 28455 ax-hfvadd 28456 ax-hvcom 28457 ax-hvass 28458 ax-hv0cl 28459 ax-hvaddid 28460 ax-hfvmul 28461 ax-hvmulid 28462 ax-hvmulass 28463 ax-hvdistr1 28464 ax-hvdistr2 28465 ax-hvmul0 28466 ax-hfi 28535 ax-his1 28538 ax-his2 28539 ax-his3 28540 ax-his4 28541 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-sup 8742 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-n0 11735 df-z 11819 df-uz 12083 df-rp 12229 df-seq 13208 df-exp 13268 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-grpo 27949 df-gid 27950 df-ablo 28001 df-vc 28015 df-nv 28048 df-va 28051 df-0v 28054 df-hnorm 28424 df-hvsub 28427 |
This theorem is referenced by: hhshsslem2 28724 hh0oi 29359 |
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