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Mirrors > Home > HSE Home > Th. List > stle0i | Structured version Visualization version GIF version |
Description: If a state is less than or equal to 0, it is 0. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sto1.1 | ⊢ 𝐴 ∈ Cℋ |
Ref | Expression |
---|---|
stle0i | ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) ≤ 0 ↔ (𝑆‘𝐴) = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sto1.1 | . . . . . 6 ⊢ 𝐴 ∈ Cℋ | |
2 | stge0 31208 | . . . . . 6 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → 0 ≤ (𝑆‘𝐴))) | |
3 | 1, 2 | mpi 20 | . . . . 5 ⊢ (𝑆 ∈ States → 0 ≤ (𝑆‘𝐴)) |
4 | 3 | anim2i 618 | . . . 4 ⊢ (((𝑆‘𝐴) ≤ 0 ∧ 𝑆 ∈ States) → ((𝑆‘𝐴) ≤ 0 ∧ 0 ≤ (𝑆‘𝐴))) |
5 | 4 | expcom 415 | . . 3 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) ≤ 0 → ((𝑆‘𝐴) ≤ 0 ∧ 0 ≤ (𝑆‘𝐴)))) |
6 | stcl 31200 | . . . . 5 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ ℝ)) | |
7 | 1, 6 | mpi 20 | . . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ∈ ℝ) |
8 | 0re 11164 | . . . 4 ⊢ 0 ∈ ℝ | |
9 | letri3 11247 | . . . 4 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 0 ∈ ℝ) → ((𝑆‘𝐴) = 0 ↔ ((𝑆‘𝐴) ≤ 0 ∧ 0 ≤ (𝑆‘𝐴)))) | |
10 | 7, 8, 9 | sylancl 587 | . . 3 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) = 0 ↔ ((𝑆‘𝐴) ≤ 0 ∧ 0 ≤ (𝑆‘𝐴)))) |
11 | 5, 10 | sylibrd 259 | . 2 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) ≤ 0 → (𝑆‘𝐴) = 0)) |
12 | 0le0 12261 | . . 3 ⊢ 0 ≤ 0 | |
13 | breq1 5113 | . . 3 ⊢ ((𝑆‘𝐴) = 0 → ((𝑆‘𝐴) ≤ 0 ↔ 0 ≤ 0)) | |
14 | 12, 13 | mpbiri 258 | . 2 ⊢ ((𝑆‘𝐴) = 0 → (𝑆‘𝐴) ≤ 0) |
15 | 11, 14 | impbid1 224 | 1 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) ≤ 0 ↔ (𝑆‘𝐴) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5110 ‘cfv 6501 ℝcr 11057 0cc0 11058 ≤ cle 11197 Cℋ cch 29913 Statescst 29946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-i2m1 11126 ax-1ne0 11127 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-hilex 29983 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-icc 13278 df-sh 30191 df-ch 30205 df-st 31195 |
This theorem is referenced by: (None) |
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