![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > stcltr2i | Structured version Visualization version GIF version |
Description: Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
stcltr1.1 | ⊢ (𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦))) |
stcltr1.2 | ⊢ 𝐴 ∈ Cℋ |
Ref | Expression |
---|---|
stcltr2i | ⊢ (𝜑 → ((𝑆‘𝐴) = 1 → 𝐴 = ℋ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . . 3 ⊢ ((𝑆‘𝐴) = 1 → ((𝑆‘ ℋ) = 1 → (𝑆‘𝐴) = 1)) | |
2 | stcltr1.1 | . . . 4 ⊢ (𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦))) | |
3 | helch 31005 | . . . 4 ⊢ ℋ ∈ Cℋ | |
4 | stcltr1.2 | . . . 4 ⊢ 𝐴 ∈ Cℋ | |
5 | 2, 3, 4 | stcltr1i 32036 | . . 3 ⊢ (𝜑 → (((𝑆‘ ℋ) = 1 → (𝑆‘𝐴) = 1) → ℋ ⊆ 𝐴)) |
6 | 1, 5 | syl5 34 | . 2 ⊢ (𝜑 → ((𝑆‘𝐴) = 1 → ℋ ⊆ 𝐴)) |
7 | 4 | chssii 30993 | . . 3 ⊢ 𝐴 ⊆ ℋ |
8 | eqss 3992 | . . 3 ⊢ (𝐴 = ℋ ↔ (𝐴 ⊆ ℋ ∧ ℋ ⊆ 𝐴)) | |
9 | 7, 8 | mpbiran 706 | . 2 ⊢ (𝐴 = ℋ ↔ ℋ ⊆ 𝐴) |
10 | 6, 9 | imbitrrdi 251 | 1 ⊢ (𝜑 → ((𝑆‘𝐴) = 1 → 𝐴 = ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ⊆ wss 3943 ‘cfv 6537 1c1 11113 ℋchba 30681 Cℋ cch 30691 Statescst 30724 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-1cn 11170 ax-addcl 11172 ax-hilex 30761 ax-hfvadd 30762 ax-hv0cl 30765 ax-hfvmul 30767 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-map 8824 df-nn 12217 df-hlim 30734 df-sh 30969 df-ch 30983 |
This theorem is referenced by: stcltrlem1 32038 |
Copyright terms: Public domain | W3C validator |