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Mirrors > Home > HSE Home > Th. List > stge0 | Structured version Visualization version GIF version |
Description: The value of a state is nonnegative. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
stge0 | ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → 0 ≤ (𝑆‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sticl 29074 | . 2 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ (0[,]1))) | |
2 | 0re 10040 | . . . 4 ⊢ 0 ∈ ℝ | |
3 | 1re 10039 | . . . 4 ⊢ 1 ∈ ℝ | |
4 | 2, 3 | elicc2i 12239 | . . 3 ⊢ ((𝑆‘𝐴) ∈ (0[,]1) ↔ ((𝑆‘𝐴) ∈ ℝ ∧ 0 ≤ (𝑆‘𝐴) ∧ (𝑆‘𝐴) ≤ 1)) |
5 | 4 | simp2bi 1077 | . 2 ⊢ ((𝑆‘𝐴) ∈ (0[,]1) → 0 ≤ (𝑆‘𝐴)) |
6 | 1, 5 | syl6 35 | 1 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → 0 ≤ (𝑆‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 ≤ cle 10075 [,]cicc 12178 Cℋ cch 27786 Statescst 27819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-hilex 27856 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-icc 12182 df-sh 28064 df-ch 28078 df-st 29070 |
This theorem is referenced by: stle0i 29098 |
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