Theorem staddi 30025

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 29995	. . . . . . 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 29995	. . . . . . 7 ⊢ (𝑆 ∈ States → (𝐵 ∈ Cℋ → (𝑆‘𝐵) ∈ ℝ))
6	4, 5	mpi 20	. . . . . 6 ⊢ (𝑆 ∈ States → (𝑆‘𝐵) ∈ ℝ)
7	3, 6	readdcld 10672	. . . . 5 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + (𝑆‘𝐵)) ∈ ℝ)
8		ltne 10739	. . . . . 6 ⊢ ((((𝑆‘𝐴) + (𝑆‘𝐵)) ∈ ℝ ∧ ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2) → 2 ≠ ((𝑆‘𝐴) + (𝑆‘𝐵)))
9	8	necomd 3073	. . . . 5 ⊢ ((((𝑆‘𝐴) + (𝑆‘𝐵)) ∈ ℝ ∧ ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2) → ((𝑆‘𝐴) + (𝑆‘𝐵)) ≠ 2)
10	7, 9	sylan 582	. . . 4 ⊢ ((𝑆 ∈ States ∧ ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2) → ((𝑆‘𝐴) + (𝑆‘𝐵)) ≠ 2)
11	10	ex 415	. . 3 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + (𝑆‘𝐵)) < 2 → ((𝑆‘𝐴) + (𝑆‘𝐵)) ≠ 2))
12	11	necon2bd 3034	. 2 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + (𝑆‘𝐵)) = 2 → ¬ ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2))
13		1re 10643	. . . . . . . . 9 ⊢ 1 ∈ ℝ
14	13	a1i 11	. . . . . . . 8 ⊢ (𝑆 ∈ States → 1 ∈ ℝ)
15		stle1 30004	. . . . . . . . 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 11257	. . . . . . 7 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + (𝑆‘𝐵)) ≤ ((𝑆‘𝐴) + 1))
18	17	adantr 483	. . . . . 6 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + (𝑆‘𝐵)) ≤ ((𝑆‘𝐴) + 1))
19		ltadd1 11109	. . . . . . . . 9 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) < 1 ↔ ((𝑆‘𝐴) + 1) < (1 + 1)))
20	19	biimpd 231	. . . . . . . 8 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) < 1 → ((𝑆‘𝐴) + 1) < (1 + 1)))
21	3, 14, 14, 20	syl3anc 1367	. . . . . . 7 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) < 1 → ((𝑆‘𝐴) + 1) < (1 + 1)))
22	21	imp 409	. . . . . 6 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + 1) < (1 + 1))
23		readdcl 10622	. . . . . . . . 9 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) + 1) ∈ ℝ)
24	3, 13, 23	sylancl 588	. . . . . . . 8 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + 1) ∈ ℝ)
25	13, 13	readdcli 10658	. . . . . . . . 9 ⊢ (1 + 1) ∈ ℝ
26	25	a1i 11	. . . . . . . 8 ⊢ (𝑆 ∈ States → (1 + 1) ∈ ℝ)
27		lelttr 10733	. . . . . . . 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 1367	. . . . . . 7 ⊢ (𝑆 ∈ States → ((((𝑆‘𝐴) + (𝑆‘𝐵)) ≤ ((𝑆‘𝐴) + 1) ∧ ((𝑆‘𝐴) + 1) < (1 + 1)) → ((𝑆‘𝐴) + (𝑆‘𝐵)) < (1 + 1)))
29	28	adantr 483	. . . . . 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 697	. . . . 5 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + (𝑆‘𝐵)) < (1 + 1))
31		df-2 11703	. . . . 5 ⊢ 2 = (1 + 1)
32	30, 31	breqtrrdi 5110	. . . 4 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2)
33	32	ex 415	. . 3 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) < 1 → ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2))
34	33	con3d 155	. 2 ⊢ (𝑆 ∈ States → (¬ ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2 → ¬ (𝑆‘𝐴) < 1))
35		stle1 30004	. . . . 5 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ≤ 1))
36	1, 35	mpi 20	. . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ≤ 1)
37		leloe 10729	. . . . 5 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) ≤ 1 ↔ ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1)))
38	3, 13, 37	sylancl 588	. . . 4 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) ≤ 1 ↔ ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1)))
39	36, 38	mpbid 234	. . 3 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1))
40	39	ord 860	. 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 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018   class class class wbr 5068  ‘cfv 6357  (class class class)co 7158  ℝcr 10538  1c1 10540   + caddc 10542   < clt 10677   ≤ cle 10678  2c2 11695   C_ℋ cch 28708  Statescst 28741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-hilex 28778

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-po 5476  df-so 5477  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-er 8291  df-map 8410  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-2 11703  df-icc 12748  df-sh 28986  df-ch 29000  df-st 29990

This theorem is referenced by: (None)