Theorem stadd3i 32267

Description: If the sum of 3 states is 3, then each state is 1. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
stle.1	⊢ 𝐴 ∈ Cℋ
stle.2	⊢ 𝐵 ∈ Cℋ
stm1add3.3	⊢ 𝐶 ∈ Cℋ

Assertion

Ref	Expression
stadd3i	⊢ (𝑆 ∈ States → ((((𝑆‘𝐴) + (𝑆‘𝐵)) + (𝑆‘𝐶)) = 3 → (𝑆‘𝐴) = 1))

Proof of Theorem stadd3i

Step	Hyp	Ref	Expression
1		stle.1	. . . . . 6 ⊢ 𝐴 ∈ Cℋ
2		stcl 32235	. . . . . 6 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ ℝ))
3	1, 2	mpi 20	. . . . 5 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ∈ ℝ)
4	3	recnd 11289	. . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ∈ ℂ)
5		stle.2	. . . . . 6 ⊢ 𝐵 ∈ Cℋ
6		stcl 32235	. . . . . 6 ⊢ (𝑆 ∈ States → (𝐵 ∈ Cℋ → (𝑆‘𝐵) ∈ ℝ))
7	5, 6	mpi 20	. . . . 5 ⊢ (𝑆 ∈ States → (𝑆‘𝐵) ∈ ℝ)
8	7	recnd 11289	. . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐵) ∈ ℂ)
9		stm1add3.3	. . . . . 6 ⊢ 𝐶 ∈ Cℋ
10		stcl 32235	. . . . . 6 ⊢ (𝑆 ∈ States → (𝐶 ∈ Cℋ → (𝑆‘𝐶) ∈ ℝ))
11	9, 10	mpi 20	. . . . 5 ⊢ (𝑆 ∈ States → (𝑆‘𝐶) ∈ ℝ)
12	11	recnd 11289	. . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐶) ∈ ℂ)
13	4, 8, 12	addassd 11283	. . 3 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + (𝑆‘𝐵)) + (𝑆‘𝐶)) = ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))))
14	13	eqeq1d 2739	. 2 ⊢ (𝑆 ∈ States → ((((𝑆‘𝐴) + (𝑆‘𝐵)) + (𝑆‘𝐶)) = 3 ↔ ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) = 3))
15		eqcom 2744	. . . 4 ⊢ (((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) = 3 ↔ 3 = ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))))
16	7, 11	readdcld 11290	. . . . . . 7 ⊢ (𝑆 ∈ States → ((𝑆‘𝐵) + (𝑆‘𝐶)) ∈ ℝ)
17	3, 16	readdcld 11290	. . . . . 6 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) ∈ ℝ)
18		ltne 11358	. . . . . . 7 ⊢ ((((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) ∈ ℝ ∧ ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < 3) → 3 ≠ ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))))
19	18	ex 412	. . . . . 6 ⊢ (((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) ∈ ℝ → (((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < 3 → 3 ≠ ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶)))))
20	17, 19	syl 17	. . . . 5 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < 3 → 3 ≠ ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶)))))
21	20	necon2bd 2956	. . . 4 ⊢ (𝑆 ∈ States → (3 = ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) → ¬ ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < 3))
22	15, 21	biimtrid 242	. . 3 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) = 3 → ¬ ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < 3))
23		1re 11261	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
24	23, 23	readdcli 11276	. . . . . . . . . 10 ⊢ (1 + 1) ∈ ℝ
25	24	a1i 11	. . . . . . . . 9 ⊢ (𝑆 ∈ States → (1 + 1) ∈ ℝ)
26		1red 11262	. . . . . . . . . 10 ⊢ (𝑆 ∈ States → 1 ∈ ℝ)
27		stle1 32244	. . . . . . . . . . 11 ⊢ (𝑆 ∈ States → (𝐵 ∈ Cℋ → (𝑆‘𝐵) ≤ 1))
28	5, 27	mpi 20	. . . . . . . . . 10 ⊢ (𝑆 ∈ States → (𝑆‘𝐵) ≤ 1)
29		stle1 32244	. . . . . . . . . . 11 ⊢ (𝑆 ∈ States → (𝐶 ∈ Cℋ → (𝑆‘𝐶) ≤ 1))
30	9, 29	mpi 20	. . . . . . . . . 10 ⊢ (𝑆 ∈ States → (𝑆‘𝐶) ≤ 1)
31	7, 11, 26, 26, 28, 30	le2addd 11882	. . . . . . . . 9 ⊢ (𝑆 ∈ States → ((𝑆‘𝐵) + (𝑆‘𝐶)) ≤ (1 + 1))
32	16, 25, 3, 31	leadd2dd 11878	. . . . . . . 8 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) ≤ ((𝑆‘𝐴) + (1 + 1)))
33	32	adantr 480	. . . . . . 7 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) ≤ ((𝑆‘𝐴) + (1 + 1)))
34		ltadd1 11730	. . . . . . . . . 10 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (1 + 1) ∈ ℝ) → ((𝑆‘𝐴) < 1 ↔ ((𝑆‘𝐴) + (1 + 1)) < (1 + (1 + 1))))
35	34	biimpd 229	. . . . . . . . 9 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (1 + 1) ∈ ℝ) → ((𝑆‘𝐴) < 1 → ((𝑆‘𝐴) + (1 + 1)) < (1 + (1 + 1))))
36	3, 26, 25, 35	syl3anc 1373	. . . . . . . 8 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) < 1 → ((𝑆‘𝐴) + (1 + 1)) < (1 + (1 + 1))))
37	36	imp 406	. . . . . . 7 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + (1 + 1)) < (1 + (1 + 1)))
38		readdcl 11238	. . . . . . . . . 10 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ (1 + 1) ∈ ℝ) → ((𝑆‘𝐴) + (1 + 1)) ∈ ℝ)
39	3, 24, 38	sylancl 586	. . . . . . . . 9 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + (1 + 1)) ∈ ℝ)
40	23, 24	readdcli 11276	. . . . . . . . . 10 ⊢ (1 + (1 + 1)) ∈ ℝ
41	40	a1i 11	. . . . . . . . 9 ⊢ (𝑆 ∈ States → (1 + (1 + 1)) ∈ ℝ)
42		lelttr 11351	. . . . . . . . 9 ⊢ ((((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) ∈ ℝ ∧ ((𝑆‘𝐴) + (1 + 1)) ∈ ℝ ∧ (1 + (1 + 1)) ∈ ℝ) → ((((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) ≤ ((𝑆‘𝐴) + (1 + 1)) ∧ ((𝑆‘𝐴) + (1 + 1)) < (1 + (1 + 1))) → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < (1 + (1 + 1))))
43	17, 39, 41, 42	syl3anc 1373	. . . . . . . 8 ⊢ (𝑆 ∈ States → ((((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) ≤ ((𝑆‘𝐴) + (1 + 1)) ∧ ((𝑆‘𝐴) + (1 + 1)) < (1 + (1 + 1))) → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < (1 + (1 + 1))))
44	43	adantr 480	. . . . . . 7 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) ≤ ((𝑆‘𝐴) + (1 + 1)) ∧ ((𝑆‘𝐴) + (1 + 1)) < (1 + (1 + 1))) → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < (1 + (1 + 1))))
45	33, 37, 44	mp2and 699	. . . . . 6 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < (1 + (1 + 1)))
46		df-3 12330	. . . . . . 7 ⊢ 3 = (2 + 1)
47		df-2 12329	. . . . . . . 8 ⊢ 2 = (1 + 1)
48	47	oveq1i 7441	. . . . . . 7 ⊢ (2 + 1) = ((1 + 1) + 1)
49		ax-1cn 11213	. . . . . . . 8 ⊢ 1 ∈ ℂ
50	49, 49, 49	addassi 11271	. . . . . . 7 ⊢ ((1 + 1) + 1) = (1 + (1 + 1))
51	46, 48, 50	3eqtrri 2770	. . . . . 6 ⊢ (1 + (1 + 1)) = 3
52	45, 51	breqtrdi 5184	. . . . 5 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < 3)
53	52	ex 412	. . . 4 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) < 1 → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < 3))
54	53	con3d 152	. . 3 ⊢ (𝑆 ∈ States → (¬ ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < 3 → ¬ (𝑆‘𝐴) < 1))
55		stle1 32244	. . . . . 6 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ≤ 1))
56	1, 55	mpi 20	. . . . 5 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ≤ 1)
57		leloe 11347	. . . . . 6 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) ≤ 1 ↔ ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1)))
58	3, 23, 57	sylancl 586	. . . . 5 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) ≤ 1 ↔ ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1)))
59	56, 58	mpbid 232	. . . 4 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1))
60	59	ord 865	. . 3 ⊢ (𝑆 ∈ States → (¬ (𝑆‘𝐴) < 1 → (𝑆‘𝐴) = 1))
61	22, 54, 60	3syld 60	. 2 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) = 3 → (𝑆‘𝐴) = 1))
62	14, 61	sylbid 240	1 ⊢ (𝑆 ∈ States → ((((𝑆‘𝐴) + (𝑆‘𝐵)) + (𝑆‘𝐶)) = 3 → (𝑆‘𝐴) = 1))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  ℝcr 11154  1c1 11156   + caddc 11158   < clt 11295   ≤ cle 11296  2c2 12321  3c3 12322   C_ℋ cch 30948  Statescst 30981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-hilex 31018

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-2 12329  df-3 12330  df-icc 13394  df-sh 31226  df-ch 31240  df-st 32230

This theorem is referenced by:  golem2  32291