Theorem stge1i 28481

Description: If a state is greater than or equal to 1, it is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sto1.1	⊢ 𝐴 ∈ Cℋ

Assertion

Ref	Expression
stge1i	⊢ (𝑆 ∈ States → (1 ≤ (𝑆‘𝐴) ↔ (𝑆‘𝐴) = 1))

Proof of Theorem stge1i

Step	Hyp	Ref	Expression
1		sto1.1	. . . . . 6 ⊢ 𝐴 ∈ Cℋ
2		stle1 28468	. . . . . 6 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ≤ 1))
3	1, 2	mpi 20	. . . . 5 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ≤ 1)
4	3	anim1i 590	. . . 4 ⊢ ((𝑆 ∈ States ∧ 1 ≤ (𝑆‘𝐴)) → ((𝑆‘𝐴) ≤ 1 ∧ 1 ≤ (𝑆‘𝐴)))
5	4	ex 449	. . 3 ⊢ (𝑆 ∈ States → (1 ≤ (𝑆‘𝐴) → ((𝑆‘𝐴) ≤ 1 ∧ 1 ≤ (𝑆‘𝐴))))
6		stcl 28459	. . . . 5 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ ℝ))
7	1, 6	mpi 20	. . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ∈ ℝ)
8		1re 9918	. . . 4 ⊢ 1 ∈ ℝ
9		letri3 10002	. . . 4 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) = 1 ↔ ((𝑆‘𝐴) ≤ 1 ∧ 1 ≤ (𝑆‘𝐴))))
10	7, 8, 9	sylancl 693	. . 3 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) = 1 ↔ ((𝑆‘𝐴) ≤ 1 ∧ 1 ≤ (𝑆‘𝐴))))
11	5, 10	sylibrd 248	. 2 ⊢ (𝑆 ∈ States → (1 ≤ (𝑆‘𝐴) → (𝑆‘𝐴) = 1))
12		1le1 10534	. . 3 ⊢ 1 ≤ 1
13		breq2 4587	. . 3 ⊢ ((𝑆‘𝐴) = 1 → (1 ≤ (𝑆‘𝐴) ↔ 1 ≤ 1))
14	12, 13	mpbiri 247	. 2 ⊢ ((𝑆‘𝐴) = 1 → 1 ≤ (𝑆‘𝐴))
15	11, 14	impbid1 214	1 ⊢ (𝑆 ∈ States → (1 ≤ (𝑆‘𝐴) ↔ (𝑆‘𝐴) = 1))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  ℝcr 9814  1c1 9816   ≤ cle 9954   C_ℋ cch 27170  Statescst 27203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-hilex 27240

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-icc 12053  df-sh 27448  df-ch 27462  df-st 28454

This theorem is referenced by:  stm1i  28486