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Theorem staddi 28295
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 𝐴C
stle.2 𝐵C
Assertion
Ref Expression
staddi (𝑆 ∈ States → (((𝑆𝐴) + (𝑆𝐵)) = 2 → (𝑆𝐴) = 1))

Proof of Theorem staddi
StepHypRef Expression
1 stle.1 . . . . . . 7 𝐴C
2 stcl 28265 . . . . . . 7 (𝑆 ∈ States → (𝐴C → (𝑆𝐴) ∈ ℝ))
31, 2mpi 20 . . . . . 6 (𝑆 ∈ States → (𝑆𝐴) ∈ ℝ)
4 stle.2 . . . . . . 7 𝐵C
5 stcl 28265 . . . . . . 7 (𝑆 ∈ States → (𝐵C → (𝑆𝐵) ∈ ℝ))
64, 5mpi 20 . . . . . 6 (𝑆 ∈ States → (𝑆𝐵) ∈ ℝ)
73, 6readdcld 9925 . . . . 5 (𝑆 ∈ States → ((𝑆𝐴) + (𝑆𝐵)) ∈ ℝ)
8 ltne 9985 . . . . . 6 ((((𝑆𝐴) + (𝑆𝐵)) ∈ ℝ ∧ ((𝑆𝐴) + (𝑆𝐵)) < 2) → 2 ≠ ((𝑆𝐴) + (𝑆𝐵)))
98necomd 2836 . . . . 5 ((((𝑆𝐴) + (𝑆𝐵)) ∈ ℝ ∧ ((𝑆𝐴) + (𝑆𝐵)) < 2) → ((𝑆𝐴) + (𝑆𝐵)) ≠ 2)
107, 9sylan 486 . . . 4 ((𝑆 ∈ States ∧ ((𝑆𝐴) + (𝑆𝐵)) < 2) → ((𝑆𝐴) + (𝑆𝐵)) ≠ 2)
1110ex 448 . . 3 (𝑆 ∈ States → (((𝑆𝐴) + (𝑆𝐵)) < 2 → ((𝑆𝐴) + (𝑆𝐵)) ≠ 2))
1211necon2bd 2797 . 2 (𝑆 ∈ States → (((𝑆𝐴) + (𝑆𝐵)) = 2 → ¬ ((𝑆𝐴) + (𝑆𝐵)) < 2))
13 1re 9895 . . . . . . . . 9 1 ∈ ℝ
1413a1i 11 . . . . . . . 8 (𝑆 ∈ States → 1 ∈ ℝ)
15 stle1 28274 . . . . . . . . 9 (𝑆 ∈ States → (𝐵C → (𝑆𝐵) ≤ 1))
164, 15mpi 20 . . . . . . . 8 (𝑆 ∈ States → (𝑆𝐵) ≤ 1)
176, 14, 3, 16leadd2dd 10491 . . . . . . 7 (𝑆 ∈ States → ((𝑆𝐴) + (𝑆𝐵)) ≤ ((𝑆𝐴) + 1))
1817adantr 479 . . . . . 6 ((𝑆 ∈ States ∧ (𝑆𝐴) < 1) → ((𝑆𝐴) + (𝑆𝐵)) ≤ ((𝑆𝐴) + 1))
19 ltadd1 10344 . . . . . . . . 9 (((𝑆𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆𝐴) < 1 ↔ ((𝑆𝐴) + 1) < (1 + 1)))
2019biimpd 217 . . . . . . . 8 (((𝑆𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆𝐴) < 1 → ((𝑆𝐴) + 1) < (1 + 1)))
213, 14, 14, 20syl3anc 1317 . . . . . . 7 (𝑆 ∈ States → ((𝑆𝐴) < 1 → ((𝑆𝐴) + 1) < (1 + 1)))
2221imp 443 . . . . . 6 ((𝑆 ∈ States ∧ (𝑆𝐴) < 1) → ((𝑆𝐴) + 1) < (1 + 1))
23 readdcl 9875 . . . . . . . . 9 (((𝑆𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆𝐴) + 1) ∈ ℝ)
243, 13, 23sylancl 692 . . . . . . . 8 (𝑆 ∈ States → ((𝑆𝐴) + 1) ∈ ℝ)
2513, 13readdcli 9909 . . . . . . . . 9 (1 + 1) ∈ ℝ
2625a1i 11 . . . . . . . 8 (𝑆 ∈ States → (1 + 1) ∈ ℝ)
27 lelttr 9979 . . . . . . . 8 ((((𝑆𝐴) + (𝑆𝐵)) ∈ ℝ ∧ ((𝑆𝐴) + 1) ∈ ℝ ∧ (1 + 1) ∈ ℝ) → ((((𝑆𝐴) + (𝑆𝐵)) ≤ ((𝑆𝐴) + 1) ∧ ((𝑆𝐴) + 1) < (1 + 1)) → ((𝑆𝐴) + (𝑆𝐵)) < (1 + 1)))
287, 24, 26, 27syl3anc 1317 . . . . . . 7 (𝑆 ∈ States → ((((𝑆𝐴) + (𝑆𝐵)) ≤ ((𝑆𝐴) + 1) ∧ ((𝑆𝐴) + 1) < (1 + 1)) → ((𝑆𝐴) + (𝑆𝐵)) < (1 + 1)))
2928adantr 479 . . . . . 6 ((𝑆 ∈ States ∧ (𝑆𝐴) < 1) → ((((𝑆𝐴) + (𝑆𝐵)) ≤ ((𝑆𝐴) + 1) ∧ ((𝑆𝐴) + 1) < (1 + 1)) → ((𝑆𝐴) + (𝑆𝐵)) < (1 + 1)))
3018, 22, 29mp2and 710 . . . . 5 ((𝑆 ∈ States ∧ (𝑆𝐴) < 1) → ((𝑆𝐴) + (𝑆𝐵)) < (1 + 1))
31 df-2 10926 . . . . 5 2 = (1 + 1)
3230, 31syl6breqr 4619 . . . 4 ((𝑆 ∈ States ∧ (𝑆𝐴) < 1) → ((𝑆𝐴) + (𝑆𝐵)) < 2)
3332ex 448 . . 3 (𝑆 ∈ States → ((𝑆𝐴) < 1 → ((𝑆𝐴) + (𝑆𝐵)) < 2))
3433con3d 146 . 2 (𝑆 ∈ States → (¬ ((𝑆𝐴) + (𝑆𝐵)) < 2 → ¬ (𝑆𝐴) < 1))
35 stle1 28274 . . . . 5 (𝑆 ∈ States → (𝐴C → (𝑆𝐴) ≤ 1))
361, 35mpi 20 . . . 4 (𝑆 ∈ States → (𝑆𝐴) ≤ 1)
37 leloe 9975 . . . . 5 (((𝑆𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑆𝐴) ≤ 1 ↔ ((𝑆𝐴) < 1 ∨ (𝑆𝐴) = 1)))
383, 13, 37sylancl 692 . . . 4 (𝑆 ∈ States → ((𝑆𝐴) ≤ 1 ↔ ((𝑆𝐴) < 1 ∨ (𝑆𝐴) = 1)))
3936, 38mpbid 220 . . 3 (𝑆 ∈ States → ((𝑆𝐴) < 1 ∨ (𝑆𝐴) = 1))
4039ord 390 . 2 (𝑆 ∈ States → (¬ (𝑆𝐴) < 1 → (𝑆𝐴) = 1))
4112, 34, 403syld 57 1 (𝑆 ∈ States → (((𝑆𝐴) + (𝑆𝐵)) = 2 → (𝑆𝐴) = 1))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779   class class class wbr 4577  cfv 5790  (class class class)co 6527  cr 9791  1c1 9793   + caddc 9795   < clt 9930  cle 9931  2c2 10917   C cch 26976  Statescst 27009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-hilex 27046
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-po 4949  df-so 4950  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-2 10926  df-icc 12009  df-sh 27254  df-ch 27268  df-st 28260
This theorem is referenced by: (None)
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