Theorem staddi 32275

Description: If the sum of 2 states is 2, then each state is 1. (Contributed by NM, 12-Nov-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
stle.1	⊢ 𝐴 ∈ Cℋ
stle.2	⊢ 𝐵 ∈ Cℋ

Assertion

Ref	Expression
staddi	⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + (𝑆‘𝐵)) = 2 → (𝑆‘𝐴) = 1))

Proof of Theorem staddi

Step	Hyp	Ref	Expression
1		stle.1	. . . . . . 7 ⊢ 𝐴 ∈ Cℋ
2		stcl 32245	. . . . . . 7 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ ℝ))
3	1, 2	mpi 20	. . . . . 6 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ∈ ℝ)
4		stle.2	. . . . . . 7 ⊢ 𝐵 ∈ Cℋ
5		stcl 32245	. . . . . . 7 ⊢ (𝑆 ∈ States → (𝐵 ∈ Cℋ → (𝑆‘𝐵) ∈ ℝ))
6	4, 5	mpi 20	. . . . . 6 ⊢ (𝑆 ∈ States → (𝑆‘𝐵) ∈ ℝ)
7	3, 6	readdcld 11288	. . . . 5 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + (𝑆‘𝐵)) ∈ ℝ)
8		ltne 11356	. . . . . 6 ⊢ ((((𝑆‘𝐴) + (𝑆‘𝐵)) ∈ ℝ ∧ ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2) → 2 ≠ ((𝑆‘𝐴) + (𝑆‘𝐵)))
9	8	necomd 2994	. . . . 5 ⊢ ((((𝑆‘𝐴) + (𝑆‘𝐵)) ∈ ℝ ∧ ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2) → ((𝑆‘𝐴) + (𝑆‘𝐵)) ≠ 2)
10	7, 9	sylan 580	. . . 4 ⊢ ((𝑆 ∈ States ∧ ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2) → ((𝑆‘𝐴) + (𝑆‘𝐵)) ≠ 2)
11	10	ex 412	. . 3 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + (𝑆‘𝐵)) < 2 → ((𝑆‘𝐴) + (𝑆‘𝐵)) ≠ 2))
12	11	necon2bd 2954	. 2 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + (𝑆‘𝐵)) = 2 → ¬ ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2))
13		1re 11259	. . . . . . . . 9 ⊢ 1 ∈ ℝ
14	13	a1i 11	. . . . . . . 8 ⊢ (𝑆 ∈ States → 1 ∈ ℝ)
15		stle1 32254	. . . . . . . . 9 ⊢ (𝑆 ∈ States → (𝐵 ∈ Cℋ → (𝑆‘𝐵) ≤ 1))
16	4, 15	mpi 20	. . . . . . . 8 ⊢ (𝑆 ∈ States → (𝑆‘𝐵) ≤ 1)
17	6, 14, 3, 16	leadd2dd 11876	. . . . . . 7 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + (𝑆‘𝐵)) ≤ ((𝑆‘𝐴) + 1))
18	17	adantr 480	. . . . . 6 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + (𝑆‘𝐵)) ≤ ((𝑆‘𝐴) + 1))
19		ltadd1 11728	. . . . . . . . 9 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) < 1 ↔ ((𝑆‘𝐴) + 1) < (1 + 1)))
20	19	biimpd 229	. . . . . . . 8 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) < 1 → ((𝑆‘𝐴) + 1) < (1 + 1)))
21	3, 14, 14, 20	syl3anc 1370	. . . . . . 7 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) < 1 → ((𝑆‘𝐴) + 1) < (1 + 1)))
22	21	imp 406	. . . . . 6 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + 1) < (1 + 1))
23		readdcl 11236	. . . . . . . . 9 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) + 1) ∈ ℝ)
24	3, 13, 23	sylancl 586	. . . . . . . 8 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + 1) ∈ ℝ)
25	13, 13	readdcli 11274	. . . . . . . . 9 ⊢ (1 + 1) ∈ ℝ
26	25	a1i 11	. . . . . . . 8 ⊢ (𝑆 ∈ States → (1 + 1) ∈ ℝ)
27		lelttr 11349	. . . . . . . 8 ⊢ ((((𝑆‘𝐴) + (𝑆‘𝐵)) ∈ ℝ ∧ ((𝑆‘𝐴) + 1) ∈ ℝ ∧ (1 + 1) ∈ ℝ) → ((((𝑆‘𝐴) + (𝑆‘𝐵)) ≤ ((𝑆‘𝐴) + 1) ∧ ((𝑆‘𝐴) + 1) < (1 + 1)) → ((𝑆‘𝐴) + (𝑆‘𝐵)) < (1 + 1)))
28	7, 24, 26, 27	syl3anc 1370	. . . . . . 7 ⊢ (𝑆 ∈ States → ((((𝑆‘𝐴) + (𝑆‘𝐵)) ≤ ((𝑆‘𝐴) + 1) ∧ ((𝑆‘𝐴) + 1) < (1 + 1)) → ((𝑆‘𝐴) + (𝑆‘𝐵)) < (1 + 1)))
29	28	adantr 480	. . . . . 6 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((((𝑆‘𝐴) + (𝑆‘𝐵)) ≤ ((𝑆‘𝐴) + 1) ∧ ((𝑆‘𝐴) + 1) < (1 + 1)) → ((𝑆‘𝐴) + (𝑆‘𝐵)) < (1 + 1)))
30	18, 22, 29	mp2and 699	. . . . 5 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + (𝑆‘𝐵)) < (1 + 1))
31		df-2 12327	. . . . 5 ⊢ 2 = (1 + 1)
32	30, 31	breqtrrdi 5190	. . . 4 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2)
33	32	ex 412	. . 3 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) < 1 → ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2))
34	33	con3d 152	. 2 ⊢ (𝑆 ∈ States → (¬ ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2 → ¬ (𝑆‘𝐴) < 1))
35		stle1 32254	. . . . 5 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ≤ 1))
36	1, 35	mpi 20	. . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ≤ 1)
37		leloe 11345	. . . . 5 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) ≤ 1 ↔ ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1)))
38	3, 13, 37	sylancl 586	. . . 4 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) ≤ 1 ↔ ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1)))
39	36, 38	mpbid 232	. . 3 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1))
40	39	ord 864	. 2 ⊢ (𝑆 ∈ States → (¬ (𝑆‘𝐴) < 1 → (𝑆‘𝐴) = 1))
41	12, 34, 40	3syld 60	1 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + (𝑆‘𝐵)) = 2 → (𝑆‘𝐴) = 1))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  ℝcr 11152  1c1 11154   + caddc 11156   < clt 11293   ≤ cle 11294  2c2 12319   C_ℋ cch 30958  Statescst 30991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-hilex 31028

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-2 12327  df-icc 13391  df-sh 31236  df-ch 31250  df-st 32240

This theorem is referenced by: (None)