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Mirrors > Home > HSE Home > Th. List > hstpyth | Structured version Visualization version GIF version |
Description: Pythagorean property of a Hilbert-space-valued state for orthogonal vectors 𝐴 and 𝐵. (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hstpyth | ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → ((normℎ‘(𝑆‘(𝐴 ∨ℋ 𝐵)))↑2) = (((normℎ‘(𝑆‘𝐴))↑2) + ((normℎ‘(𝑆‘𝐵))↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hstosum 29977 | . . . 4 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵))) | |
2 | 1 | fveq2d 6655 | . . 3 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → (normℎ‘(𝑆‘(𝐴 ∨ℋ 𝐵))) = (normℎ‘((𝑆‘𝐴) +ℎ (𝑆‘𝐵)))) |
3 | 2 | oveq1d 7152 | . 2 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → ((normℎ‘(𝑆‘(𝐴 ∨ℋ 𝐵)))↑2) = ((normℎ‘((𝑆‘𝐴) +ℎ (𝑆‘𝐵)))↑2)) |
4 | hstcl 29973 | . . . 4 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → (𝑆‘𝐴) ∈ ℋ) | |
5 | 4 | adantr 483 | . . 3 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → (𝑆‘𝐴) ∈ ℋ) |
6 | hstcl 29973 | . . . 4 ⊢ ((𝑆 ∈ CHStates ∧ 𝐵 ∈ Cℋ ) → (𝑆‘𝐵) ∈ ℋ) | |
7 | 6 | ad2ant2r 745 | . . 3 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → (𝑆‘𝐵) ∈ ℋ) |
8 | hstorth 29976 | . . 3 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → ((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0) | |
9 | normpyth 28901 | . . . 4 ⊢ (((𝑆‘𝐴) ∈ ℋ ∧ (𝑆‘𝐵) ∈ ℋ) → (((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0 → ((normℎ‘((𝑆‘𝐴) +ℎ (𝑆‘𝐵)))↑2) = (((normℎ‘(𝑆‘𝐴))↑2) + ((normℎ‘(𝑆‘𝐵))↑2)))) | |
10 | 9 | 3impia 1113 | . . 3 ⊢ (((𝑆‘𝐴) ∈ ℋ ∧ (𝑆‘𝐵) ∈ ℋ ∧ ((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0) → ((normℎ‘((𝑆‘𝐴) +ℎ (𝑆‘𝐵)))↑2) = (((normℎ‘(𝑆‘𝐴))↑2) + ((normℎ‘(𝑆‘𝐵))↑2))) |
11 | 5, 7, 8, 10 | syl3anc 1367 | . 2 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → ((normℎ‘((𝑆‘𝐴) +ℎ (𝑆‘𝐵)))↑2) = (((normℎ‘(𝑆‘𝐴))↑2) + ((normℎ‘(𝑆‘𝐵))↑2))) |
12 | 3, 11 | eqtrd 2855 | 1 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → ((normℎ‘(𝑆‘(𝐴 ∨ℋ 𝐵)))↑2) = (((normℎ‘(𝑆‘𝐴))↑2) + ((normℎ‘(𝑆‘𝐵))↑2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3919 ‘cfv 6336 (class class class)co 7137 0cc0 10518 + caddc 10521 2c2 11674 ↑cexp 13414 ℋchba 28675 +ℎ cva 28676 ·ih csp 28678 normℎcno 28679 Cℋ cch 28685 ⊥cort 28686 ∨ℋ chj 28689 CHStateschst 28719 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 ax-pre-sup 10596 ax-hilex 28755 ax-hfvadd 28756 ax-hv0cl 28759 ax-hvmul0 28766 ax-hfi 28835 ax-his1 28838 ax-his2 28839 ax-his3 28840 ax-his4 28841 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-2nd 7671 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-er 8270 df-map 8389 df-en 8491 df-dom 8492 df-sdom 8493 df-sup 8887 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-div 11279 df-nn 11620 df-2 11682 df-3 11683 df-n0 11880 df-z 11964 df-uz 12226 df-rp 12372 df-seq 13355 df-exp 13415 df-cj 14438 df-re 14439 df-im 14440 df-sqrt 14574 df-hnorm 28724 df-sh 28963 df-ch 28977 df-hst 29968 |
This theorem is referenced by: hstle 29986 |
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