Theorem staddi 32321

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 32291	. . . . . . 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 32291	. . . . . . 7 ⊢ (𝑆 ∈ States → (𝐵 ∈ Cℋ → (𝑆‘𝐵) ∈ ℝ))
6	4, 5	mpi 20	. . . . . 6 ⊢ (𝑆 ∈ States → (𝑆‘𝐵) ∈ ℝ)
7	3, 6	readdcld 11161	. . . . 5 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + (𝑆‘𝐵)) ∈ ℝ)
8		ltne 11230	. . . . . 6 ⊢ ((((𝑆‘𝐴) + (𝑆‘𝐵)) ∈ ℝ ∧ ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2) → 2 ≠ ((𝑆‘𝐴) + (𝑆‘𝐵)))
9	8	necomd 2987	. . . . 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 2948	. 2 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + (𝑆‘𝐵)) = 2 → ¬ ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2))
13		1re 11132	. . . . . . . . 9 ⊢ 1 ∈ ℝ
14	13	a1i 11	. . . . . . . 8 ⊢ (𝑆 ∈ States → 1 ∈ ℝ)
15		stle1 32300	. . . . . . . . 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 11752	. . . . . . 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 11604	. . . . . . . . 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 1373	. . . . . . 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 11109	. . . . . . . . 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 11147	. . . . . . . . 9 ⊢ (1 + 1) ∈ ℝ
26	25	a1i 11	. . . . . . . 8 ⊢ (𝑆 ∈ States → (1 + 1) ∈ ℝ)
27		lelttr 11223	. . . . . . . 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 1373	. . . . . . 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 12208	. . . . 5 ⊢ 2 = (1 + 1)
32	30, 31	breqtrrdi 5140	. . . 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 32300	. . . . 5 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ≤ 1))
36	1, 35	mpi 20	. . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ≤ 1)
37		leloe 11219	. . . . 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 1541   ∈ wcel 2113   ≠ wne 2932   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  ℝcr 11025  1c1 11027   + caddc 11029   < clt 11166   ≤ cle 11167  2c2 12200   C_ℋ cch 31004  Statescst 31037

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-hilex 31074

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-2 12208  df-icc 13268  df-sh 31282  df-ch 31296  df-st 32286

This theorem is referenced by: (None)