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Mirrors > Home > HSE Home > Th. List > stge1i | Structured version Visualization version GIF version |
Description: If a state is greater than or equal to 1, it is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sto1.1 | ⊢ 𝐴 ∈ Cℋ |
Ref | Expression |
---|---|
stge1i | ⊢ (𝑆 ∈ States → (1 ≤ (𝑆‘𝐴) ↔ (𝑆‘𝐴) = 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sto1.1 | . . . . . 6 ⊢ 𝐴 ∈ Cℋ | |
2 | stle1 32048 | . . . . . 6 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ≤ 1)) | |
3 | 1, 2 | mpi 20 | . . . . 5 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ≤ 1) |
4 | 3 | anim1i 614 | . . . 4 ⊢ ((𝑆 ∈ States ∧ 1 ≤ (𝑆‘𝐴)) → ((𝑆‘𝐴) ≤ 1 ∧ 1 ≤ (𝑆‘𝐴))) |
5 | 4 | ex 412 | . . 3 ⊢ (𝑆 ∈ States → (1 ≤ (𝑆‘𝐴) → ((𝑆‘𝐴) ≤ 1 ∧ 1 ≤ (𝑆‘𝐴)))) |
6 | stcl 32039 | . . . . 5 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ ℝ)) | |
7 | 1, 6 | mpi 20 | . . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ∈ ℝ) |
8 | 1re 11245 | . . . 4 ⊢ 1 ∈ ℝ | |
9 | letri3 11330 | . . . 4 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) = 1 ↔ ((𝑆‘𝐴) ≤ 1 ∧ 1 ≤ (𝑆‘𝐴)))) | |
10 | 7, 8, 9 | sylancl 585 | . . 3 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) = 1 ↔ ((𝑆‘𝐴) ≤ 1 ∧ 1 ≤ (𝑆‘𝐴)))) |
11 | 5, 10 | sylibrd 259 | . 2 ⊢ (𝑆 ∈ States → (1 ≤ (𝑆‘𝐴) → (𝑆‘𝐴) = 1)) |
12 | 1le1 11873 | . . 3 ⊢ 1 ≤ 1 | |
13 | breq2 5152 | . . 3 ⊢ ((𝑆‘𝐴) = 1 → (1 ≤ (𝑆‘𝐴) ↔ 1 ≤ 1)) | |
14 | 12, 13 | mpbiri 258 | . 2 ⊢ ((𝑆‘𝐴) = 1 → 1 ≤ (𝑆‘𝐴)) |
15 | 11, 14 | impbid1 224 | 1 ⊢ (𝑆 ∈ States → (1 ≤ (𝑆‘𝐴) ↔ (𝑆‘𝐴) = 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 ‘cfv 6548 ℝcr 11138 1c1 11140 ≤ cle 11280 Cℋ cch 30752 Statescst 30785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-i2m1 11207 ax-1ne0 11208 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-hilex 30822 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-icc 13364 df-sh 31030 df-ch 31044 df-st 32034 |
This theorem is referenced by: stm1i 32066 |
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