Theorem stadd3i 32071

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 32039	. . . . . 6 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ ℝ))
3	1, 2	mpi 20	. . . . 5 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ∈ ℝ)
4	3	recnd 11273	. . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ∈ ℂ)
5		stle.2	. . . . . 6 ⊢ 𝐵 ∈ Cℋ
6		stcl 32039	. . . . . 6 ⊢ (𝑆 ∈ States → (𝐵 ∈ Cℋ → (𝑆‘𝐵) ∈ ℝ))
7	5, 6	mpi 20	. . . . 5 ⊢ (𝑆 ∈ States → (𝑆‘𝐵) ∈ ℝ)
8	7	recnd 11273	. . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐵) ∈ ℂ)
9		stm1add3.3	. . . . . 6 ⊢ 𝐶 ∈ Cℋ
10		stcl 32039	. . . . . 6 ⊢ (𝑆 ∈ States → (𝐶 ∈ Cℋ → (𝑆‘𝐶) ∈ ℝ))
11	9, 10	mpi 20	. . . . 5 ⊢ (𝑆 ∈ States → (𝑆‘𝐶) ∈ ℝ)
12	11	recnd 11273	. . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐶) ∈ ℂ)
13	4, 8, 12	addassd 11267	. . 3 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + (𝑆‘𝐵)) + (𝑆‘𝐶)) = ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))))
14	13	eqeq1d 2730	. 2 ⊢ (𝑆 ∈ States → ((((𝑆‘𝐴) + (𝑆‘𝐵)) + (𝑆‘𝐶)) = 3 ↔ ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) = 3))
15		eqcom 2735	. . . 4 ⊢ (((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) = 3 ↔ 3 = ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))))
16	7, 11	readdcld 11274	. . . . . . 7 ⊢ (𝑆 ∈ States → ((𝑆‘𝐵) + (𝑆‘𝐶)) ∈ ℝ)
17	3, 16	readdcld 11274	. . . . . 6 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) ∈ ℝ)
18		ltne 11342	. . . . . . 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 2953	. . . 4 ⊢ (𝑆 ∈ States → (3 = ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) → ¬ ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < 3))
22	15, 21	biimtrid 241	. . 3 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) = 3 → ¬ ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < 3))
23		1re 11245	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
24	23, 23	readdcli 11260	. . . . . . . . . 10 ⊢ (1 + 1) ∈ ℝ
25	24	a1i 11	. . . . . . . . 9 ⊢ (𝑆 ∈ States → (1 + 1) ∈ ℝ)
26		1red 11246	. . . . . . . . . 10 ⊢ (𝑆 ∈ States → 1 ∈ ℝ)
27		stle1 32048	. . . . . . . . . . 11 ⊢ (𝑆 ∈ States → (𝐵 ∈ Cℋ → (𝑆‘𝐵) ≤ 1))
28	5, 27	mpi 20	. . . . . . . . . 10 ⊢ (𝑆 ∈ States → (𝑆‘𝐵) ≤ 1)
29		stle1 32048	. . . . . . . . . . 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 11864	. . . . . . . . 9 ⊢ (𝑆 ∈ States → ((𝑆‘𝐵) + (𝑆‘𝐶)) ≤ (1 + 1))
32	16, 25, 3, 31	leadd2dd 11860	. . . . . . . 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 11712	. . . . . . . . . 10 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (1 + 1) ∈ ℝ) → ((𝑆‘𝐴) < 1 ↔ ((𝑆‘𝐴) + (1 + 1)) < (1 + (1 + 1))))
35	34	biimpd 228	. . . . . . . . 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 1369	. . . . . . . 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 11222	. . . . . . . . . 10 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ (1 + 1) ∈ ℝ) → ((𝑆‘𝐴) + (1 + 1)) ∈ ℝ)
39	3, 24, 38	sylancl 585	. . . . . . . . 9 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + (1 + 1)) ∈ ℝ)
40	23, 24	readdcli 11260	. . . . . . . . . 10 ⊢ (1 + (1 + 1)) ∈ ℝ
41	40	a1i 11	. . . . . . . . 9 ⊢ (𝑆 ∈ States → (1 + (1 + 1)) ∈ ℝ)
42		lelttr 11335	. . . . . . . . 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 1369	. . . . . . . 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 698	. . . . . 6 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < (1 + (1 + 1)))
46		df-3 12307	. . . . . . 7 ⊢ 3 = (2 + 1)
47		df-2 12306	. . . . . . . 8 ⊢ 2 = (1 + 1)
48	47	oveq1i 7430	. . . . . . 7 ⊢ (2 + 1) = ((1 + 1) + 1)
49		ax-1cn 11197	. . . . . . . 8 ⊢ 1 ∈ ℂ
50	49, 49, 49	addassi 11255	. . . . . . 7 ⊢ ((1 + 1) + 1) = (1 + (1 + 1))
51	46, 48, 50	3eqtrri 2761	. . . . . 6 ⊢ (1 + (1 + 1)) = 3
52	45, 51	breqtrdi 5189	. . . . 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 32048	. . . . . 6 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ≤ 1))
56	1, 55	mpi 20	. . . . 5 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ≤ 1)
57		leloe 11331	. . . . . 6 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) ≤ 1 ↔ ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1)))
58	3, 23, 57	sylancl 585	. . . . 5 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) ≤ 1 ↔ ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1)))
59	56, 58	mpbid 231	. . . 4 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1))
60	59	ord 863	. . 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 239	1 ⊢ (𝑆 ∈ States → ((((𝑆‘𝐴) + (𝑆‘𝐵)) + (𝑆‘𝐶)) = 3 → (𝑆‘𝐴) = 1))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 846   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2937   class class class wbr 5148  ‘cfv 6548  (class class class)co 7420  ℝcr 11138  1c1 11140   + caddc 11142   < clt 11279   ≤ cle 11280  2c2 12298  3c3 12299   C_ℋ cch 30752  Statescst 30785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-hilex 30822

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-2 12306  df-3 12307  df-icc 13364  df-sh 31030  df-ch 31044  df-st 32034

This theorem is referenced by:  golem2  32095

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