![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > stle1 | Structured version Visualization version GIF version |
Description: The value of a state is less than or equal to one. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
stle1 | ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ≤ 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sticl 29411 | . 2 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ (0[,]1))) | |
2 | 0re 10241 | . . . 4 ⊢ 0 ∈ ℝ | |
3 | 1re 10240 | . . . 4 ⊢ 1 ∈ ℝ | |
4 | 2, 3 | elicc2i 12443 | . . 3 ⊢ ((𝑆‘𝐴) ∈ (0[,]1) ↔ ((𝑆‘𝐴) ∈ ℝ ∧ 0 ≤ (𝑆‘𝐴) ∧ (𝑆‘𝐴) ≤ 1)) |
5 | 4 | simp3bi 1141 | . 2 ⊢ ((𝑆‘𝐴) ∈ (0[,]1) → (𝑆‘𝐴) ≤ 1) |
6 | 1, 5 | syl6 35 | 1 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ≤ 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2145 class class class wbr 4786 ‘cfv 6031 (class class class)co 6792 ℝcr 10136 0cc0 10137 1c1 10138 ≤ cle 10276 [,]cicc 12382 Cℋ cch 28123 Statescst 28156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-i2m1 10205 ax-1ne0 10206 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-hilex 28193 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-er 7895 df-map 8010 df-en 8109 df-dom 8110 df-sdom 8111 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-icc 12386 df-sh 28401 df-ch 28415 df-st 29407 |
This theorem is referenced by: stge1i 29434 stlei 29436 stlesi 29437 staddi 29442 stadd3i 29444 |
Copyright terms: Public domain | W3C validator |