Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > stj | Structured version Visualization version GIF version |
Description: The value of a state on a join. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
stj | ⊢ (𝑆 ∈ States → (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐴 ⊆ (⊥‘𝐵)) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isst 30603 | . . . 4 ⊢ (𝑆 ∈ States ↔ (𝑆: Cℋ ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))))) | |
2 | 1 | simp3bi 1145 | . . 3 ⊢ (𝑆 ∈ States → ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦)))) |
3 | sseq1 3948 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝑦))) | |
4 | fvoveq1 7318 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑆‘(𝑥 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝑦))) | |
5 | fveq2 6792 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴)) | |
6 | 5 | oveq1d 7310 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) + (𝑆‘𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝑦))) |
7 | 4, 6 | eqeq12d 2749 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦)) ↔ (𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝑦)))) |
8 | 3, 7 | imbi12d 344 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))) ↔ (𝐴 ⊆ (⊥‘𝑦) → (𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝑦))))) |
9 | fveq2 6792 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵)) | |
10 | 9 | sseq2d 3955 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝐵))) |
11 | oveq2 7303 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 ∨ℋ 𝑦) = (𝐴 ∨ℋ 𝐵)) | |
12 | 11 | fveq2d 6796 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑆‘(𝐴 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝐵))) |
13 | fveq2 6792 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑆‘𝑦) = (𝑆‘𝐵)) | |
14 | 13 | oveq2d 7311 | . . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝐴) + (𝑆‘𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝐵))) |
15 | 12, 14 | eqeq12d 2749 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝑦)) ↔ (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵)))) |
16 | 10, 15 | imbi12d 344 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ⊆ (⊥‘𝑦) → (𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) + (𝑆‘𝑦))) ↔ (𝐴 ⊆ (⊥‘𝐵) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵))))) |
17 | 8, 16 | rspc2v 3572 | . . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))) → (𝐴 ⊆ (⊥‘𝐵) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵))))) |
18 | 2, 17 | syl5com 31 | . 2 ⊢ (𝑆 ∈ States → ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ (⊥‘𝐵) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵))))) |
19 | 18 | impd 410 | 1 ⊢ (𝑆 ∈ States → (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐴 ⊆ (⊥‘𝐵)) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2101 ∀wral 3059 ⊆ wss 3889 ⟶wf 6443 ‘cfv 6447 (class class class)co 7295 0cc0 10899 1c1 10900 + caddc 10902 [,]cicc 13110 ℋchba 29309 Cℋ cch 29319 ⊥cort 29320 ∨ℋ chj 29323 Statescst 29352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-hilex 29389 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3224 df-v 3436 df-sbc 3719 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-br 5078 df-opab 5140 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-fv 6455 df-ov 7298 df-oprab 7299 df-mpo 7300 df-map 8637 df-sh 29597 df-ch 29611 df-st 30601 |
This theorem is referenced by: sto1i 30626 stlei 30630 stji1i 30632 |
Copyright terms: Public domain | W3C validator |