Theorem staddi 32317

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 32287	. . . . . . 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 32287	. . . . . . 7 ⊢ (𝑆 ∈ States → (𝐵 ∈ Cℋ → (𝑆‘𝐵) ∈ ℝ))
6	4, 5	mpi 20	. . . . . 6 ⊢ (𝑆 ∈ States → (𝑆‘𝐵) ∈ ℝ)
7	3, 6	readdcld 11174	. . . . 5 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + (𝑆‘𝐵)) ∈ ℝ)
8		ltne 11243	. . . . . 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 581	. . . 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 11144	. . . . . . . . 9 ⊢ 1 ∈ ℝ
14	13	a1i 11	. . . . . . . 8 ⊢ (𝑆 ∈ States → 1 ∈ ℝ)
15		stle1 32296	. . . . . . . . 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 11765	. . . . . . 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 11617	. . . . . . . . 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 1374	. . . . . . 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 11121	. . . . . . . . 9 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) + 1) ∈ ℝ)
24	3, 13, 23	sylancl 587	. . . . . . . 8 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + 1) ∈ ℝ)
25	13, 13	readdcli 11160	. . . . . . . . 9 ⊢ (1 + 1) ∈ ℝ
26	25	a1i 11	. . . . . . . 8 ⊢ (𝑆 ∈ States → (1 + 1) ∈ ℝ)
27		lelttr 11236	. . . . . . . 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 1374	. . . . . . 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 700	. . . . 5 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + (𝑆‘𝐵)) < (1 + 1))
31		df-2 12244	. . . . 5 ⊢ 2 = (1 + 1)
32	30, 31	breqtrrdi 5127	. . . 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 32296	. . . . 5 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ≤ 1))
36	1, 35	mpi 20	. . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ≤ 1)
37		leloe 11232	. . . . 5 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) ≤ 1 ↔ ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1)))
38	3, 13, 37	sylancl 587	. . . 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 865	. 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 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  ℝcr 11037  1c1 11039   + caddc 11041   < clt 11179   ≤ cle 11180  2c2 12236   C_ℋ cch 31000  Statescst 31033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-hilex 31070

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-2 12244  df-icc 13305  df-sh 31278  df-ch 31292  df-st 32282

This theorem is referenced by: (None)