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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 29689 | . . . . . 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 413 | . . 3 ⊢ (𝑆 ∈ States → (1 ≤ (𝑆‘𝐴) → ((𝑆‘𝐴) ≤ 1 ∧ 1 ≤ (𝑆‘𝐴)))) |
6 | stcl 29680 | . . . . 5 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ ℝ)) | |
7 | 1, 6 | mpi 20 | . . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ∈ ℝ) |
8 | 1re 10494 | . . . 4 ⊢ 1 ∈ ℝ | |
9 | letri3 10579 | . . . 4 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) = 1 ↔ ((𝑆‘𝐴) ≤ 1 ∧ 1 ≤ (𝑆‘𝐴)))) | |
10 | 7, 8, 9 | sylancl 586 | . . 3 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) = 1 ↔ ((𝑆‘𝐴) ≤ 1 ∧ 1 ≤ (𝑆‘𝐴)))) |
11 | 5, 10 | sylibrd 260 | . 2 ⊢ (𝑆 ∈ States → (1 ≤ (𝑆‘𝐴) → (𝑆‘𝐴) = 1)) |
12 | 1le1 11122 | . . 3 ⊢ 1 ≤ 1 | |
13 | breq2 4972 | . . 3 ⊢ ((𝑆‘𝐴) = 1 → (1 ≤ (𝑆‘𝐴) ↔ 1 ≤ 1)) | |
14 | 12, 13 | mpbiri 259 | . 2 ⊢ ((𝑆‘𝐴) = 1 → 1 ≤ (𝑆‘𝐴)) |
15 | 11, 14 | impbid1 226 | 1 ⊢ (𝑆 ∈ States → (1 ≤ (𝑆‘𝐴) ↔ (𝑆‘𝐴) = 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1525 ∈ wcel 2083 class class class wbr 4968 ‘cfv 6232 ℝcr 10389 1c1 10391 ≤ cle 10529 Cℋ cch 28393 Statescst 28426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-i2m1 10458 ax-1ne0 10459 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-hilex 28463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-po 5369 df-so 5370 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-ov 7026 df-oprab 7027 df-mpo 7028 df-er 8146 df-map 8265 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-icc 12599 df-sh 28671 df-ch 28685 df-st 29675 |
This theorem is referenced by: stm1i 29707 |
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