Theorem staddi 32278

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 32248	. . . . . . 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 32248	. . . . . . 7 ⊢ (𝑆 ∈ States → (𝐵 ∈ Cℋ → (𝑆‘𝐵) ∈ ℝ))
6	4, 5	mpi 20	. . . . . 6 ⊢ (𝑆 ∈ States → (𝑆‘𝐵) ∈ ℝ)
7	3, 6	readdcld 11319	. . . . 5 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + (𝑆‘𝐵)) ∈ ℝ)
8		ltne 11387	. . . . . 6 ⊢ ((((𝑆‘𝐴) + (𝑆‘𝐵)) ∈ ℝ ∧ ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2) → 2 ≠ ((𝑆‘𝐴) + (𝑆‘𝐵)))
9	8	necomd 3002	. . . . 5 ⊢ ((((𝑆‘𝐴) + (𝑆‘𝐵)) ∈ ℝ ∧ ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2) → ((𝑆‘𝐴) + (𝑆‘𝐵)) ≠ 2)
10	7, 9	sylan 579	. . . 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 2962	. 2 ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + (𝑆‘𝐵)) = 2 → ¬ ((𝑆‘𝐴) + (𝑆‘𝐵)) < 2))
13		1re 11290	. . . . . . . . 9 ⊢ 1 ∈ ℝ
14	13	a1i 11	. . . . . . . 8 ⊢ (𝑆 ∈ States → 1 ∈ ℝ)
15		stle1 32257	. . . . . . . . 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 11905	. . . . . . 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 11757	. . . . . . . . 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 1371	. . . . . . 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 11267	. . . . . . . . 9 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) + 1) ∈ ℝ)
24	3, 13, 23	sylancl 585	. . . . . . . 8 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + 1) ∈ ℝ)
25	13, 13	readdcli 11305	. . . . . . . . 9 ⊢ (1 + 1) ∈ ℝ
26	25	a1i 11	. . . . . . . 8 ⊢ (𝑆 ∈ States → (1 + 1) ∈ ℝ)
27		lelttr 11380	. . . . . . . 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 1371	. . . . . . 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 698	. . . . 5 ⊢ ((𝑆 ∈ States ∧ (𝑆‘𝐴) < 1) → ((𝑆‘𝐴) + (𝑆‘𝐵)) < (1 + 1))
31		df-2 12356	. . . . 5 ⊢ 2 = (1 + 1)
32	30, 31	breqtrrdi 5208	. . . 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 32257	. . . . 5 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ≤ 1))
36	1, 35	mpi 20	. . . 4 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ≤ 1)
37		leloe 11376	. . . . 5 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆‘𝐴) ≤ 1 ↔ ((𝑆‘𝐴) < 1 ∨ (𝑆‘𝐴) = 1)))
38	3, 13, 37	sylancl 585	. . . 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 863	. 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 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  ℝcr 11183  1c1 11185   + caddc 11187   < clt 11324   ≤ cle 11325  2c2 12348   C_ℋ cch 30961  Statescst 30994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-hilex 31031

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-2 12356  df-icc 13414  df-sh 31239  df-ch 31253  df-st 32243

This theorem is referenced by: (None)