Theorem stadd3i 32843

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 32811	. . . . . 6 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ ℝ))
3	1, 2	mpi 21	. . . . 5 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ∈ ℝ)
4	3	recnd 11330	. . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ∈ ℂ)
5		stle.2	. . . . . 6 ⊢ 𝐵 ∈ Cℋ
6		stcl 32811	. . . . . 6 ⊢ (𝑆 ∈ States → (𝐵 ∈ Cℋ → (𝑆‘𝐵) ∈ ℝ))
7	5, 6	mpi 21	. . . . 5 ⊢ (𝑆 ∈ States → (𝑆‘𝐵) ∈ ℝ)
8	7	recnd 11330	. . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐵) ∈ ℂ)
9		stm1add3.3	. . . . . 6 ⊢ 𝐶 ∈ Cℋ
10		stcl 32811	. . . . . 6 ⊢ (𝑆 ∈ States → (𝐶 ∈ Cℋ → (𝑆‘𝐶) ∈ ℝ))
11	9, 10	mpi 21	. . . . 5 ⊢ (𝑆 ∈ States → (𝑆‘𝐶) ∈ ℝ)
12	11	recnd 11330	. . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐶) ∈ ℂ)
13	4, 8, 12	addassd 11324	. . 3 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + (𝑆‘𝐵)) + (𝑆‘𝐶)) = ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))))
14	13	eqeq1d 2763	. 2 ⊢ (𝑆 ∈ States → ((((𝑆‘𝐴) + (𝑆‘𝐵)) + (𝑆‘𝐶)) = 3 ↔ ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) = 3))
15		eqcom 2768	. . . 4 ⊢ (((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) = 3 ↔ 3 = ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))))
16	7, 11	readdcld 11331	. . . . . . 7 ⊢ (𝑆 ∈ States → ((𝑆‘𝐵) + (𝑆‘𝐶)) ∈ ℝ)
17	3, 16	readdcld 11331	. . . . . 6 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) ∈ ℝ)
18		ltne 11400	. . . . . . 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 418	. . . . . 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 18	. . . . 5 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < 3 → 3 ≠ ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶)))))
21	20	necon2bd 2972	. . . 4 ⊢ (𝑆 ∈ States → (3 = ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) → ¬ ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < 3))
22	15, 21	biimtrid 245	. . 3 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) = 3 → ¬ ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < 3))
23		1re 11301	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
24	23, 23	readdcli 11317	. . . . . . . . . 10 ⊢ (1 + 1) ∈ ℝ
25	24	a1i 11	. . . . . . . . 9 ⊢ (𝑆 ∈ States → (1 + 1) ∈ ℝ)
26		1red 11302	. . . . . . . . . 10 ⊢ (𝑆 ∈ States → 1 ∈ ℝ)
27		stle1 32820	. . . . . . . . . . 11 ⊢ (𝑆 ∈ States → (𝐵 ∈ Cℋ → (𝑆‘𝐵) ≤ 1))
28	5, 27	mpi 21	. . . . . . . . . 10 ⊢ (𝑆 ∈ States → (𝑆‘𝐵) ≤ 1)
29		stle1 32820	. . . . . . . . . . 11 ⊢ (𝑆 ∈ States → (𝐶 ∈ Cℋ → (𝑆‘𝐶) ≤ 1))
30	9, 29	mpi 21	. . . . . . . . . 10 ⊢ (𝑆 ∈ States → (𝑆‘𝐶) ≤ 1)
31	7, 11, 26, 26, 28, 30	le2addd 11928	. . . . . . . . 9 ⊢ (𝑆 ∈ States → ((𝑆‘𝐵) + (𝑆‘𝐶)) ≤ (1 + 1))
32	16, 25, 3, 31	leadd2dd 11924	. . . . . . . 8 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) ≤ ((𝑆‘𝐴) + (1 + 1)))
33	32	adantr 486	. . . . . . 7 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) ≤ ((𝑆‘𝐴) + (1 + 1)))
34		ltadd1 11776	. . . . . . . . . 10 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (1 + 1) ∈ ℝ) → ((𝑆‘𝐴) < 1 ↔ ((𝑆‘𝐴) + (1 + 1)) < (1 + (1 + 1))))
35	34	biimpd 232	. . . . . . . . 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 1398	. . . . . . . 8 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) < 1 → ((𝑆‘𝐴) + (1 + 1)) < (1 + (1 + 1))))
37	36	imp 412	. . . . . . 7 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + (1 + 1)) < (1 + (1 + 1)))
38		readdcl 11276	. . . . . . . . . 10 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ (1 + 1) ∈ ℝ) → ((𝑆‘𝐴) + (1 + 1)) ∈ ℝ)
39	3, 24, 38	sylancl 598	. . . . . . . . 9 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + (1 + 1)) ∈ ℝ)
40	23, 24	readdcli 11317	. . . . . . . . . 10 ⊢ (1 + (1 + 1)) ∈ ℝ
41	40	a1i 11	. . . . . . . . 9 ⊢ (𝑆 ∈ States → (1 + (1 + 1)) ∈ ℝ)
42		lelttr 11393	. . . . . . . . 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 1398	. . . . . . . 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 486	. . . . . . 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 712	. . . . . 6 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < (1 + (1 + 1)))
46		df-3 12399	. . . . . . 7 ⊢ 3 = (2 + 1)
47		df-2 12398	. . . . . . . 8 ⊢ 2 = (1 + 1)
48	47	oveq1i 7428	. . . . . . 7 ⊢ (2 + 1) = ((1 + 1) + 1)
49		ax-1cn 11251	. . . . . . . 8 ⊢ 1 ∈ ℂ
50	49, 49, 49	addassi 11312	. . . . . . 7 ⊢ ((1 + 1) + 1) = (1 + (1 + 1))
51	46, 48, 50	3eqtrri 2789	. . . . . 6 ⊢ (1 + (1 + 1)) = 3
52	45, 51	breqtrdi 5146	. . . . 5 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < 3)
53	52	ex 418	. . . 4 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) < 1 → ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < 3))
54	53	con3d 153	. . 3 ⊢ (𝑆 ∈ States → (¬ ((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) < 3 → ¬ (𝑆‘𝐴) < 1))
55		stle1 32820	. . . . . 6 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ≤ 1))
56	1, 55	mpi 21	. . . . 5 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ≤ 1)
57		leloe 11389	. . . . . 6 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) ≤ 1 ↔ ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1)))
58	3, 23, 57	sylancl 598	. . . . 5 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) ≤ 1 ↔ ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1)))
59	56, 58	mpbid 235	. . . 4 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1))
60	59	ord 878	. . 3 ⊢ (𝑆 ∈ States → (¬ (𝑆‘𝐴) < 1 → (𝑆‘𝐴) = 1))
61	22, 54, 60	3syld 61	. 2 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + ((𝑆‘𝐵) + (𝑆‘𝐶))) = 3 → (𝑆‘𝐴) = 1))
62	14, 61	sylbid 243	1 ⊢ (𝑆 ∈ States → ((((𝑆‘𝐴) + (𝑆‘𝐵)) + (𝑆‘𝐶)) = 3 → (𝑆‘𝐴) = 1))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956   class class class wbr 5103  ‘cfv 6537  (class class class)co 7418  ℝcr 11192  1c1 11194   + caddc 11196   < clt 11336   ≤ cle 11337  2c2 12390  3c3 12391   C_ℋ cch 31524  Statescst 31557

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-hilex 31594

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-po 5559  df-so 5560  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-er 8710  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-2 12398  df-3 12399  df-icc 13476  df-sh 31802  df-ch 31816  df-st 32806

This theorem is used by:  golem2  32867