HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  3oalem1 Structured version   Visualization version   GIF version

Theorem 3oalem1 28649
Description: Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
3oalem1.1 𝐵C
3oalem1.2 𝐶C
3oalem1.3 𝑅C
3oalem1.4 𝑆C
Assertion
Ref Expression
3oalem1 ((((𝑥𝐵𝑦𝑅) ∧ 𝑣 = (𝑥 + 𝑦)) ∧ ((𝑧𝐶𝑤𝑆) ∧ 𝑣 = (𝑧 + 𝑤))) → (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ) ∧ (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝐵   𝑥,𝐶,𝑦,𝑧,𝑤,𝑣   𝑥,𝑅,𝑦,𝑧,𝑤,𝑣   𝑥,𝑆,𝑦,𝑧,𝑤,𝑣

Proof of Theorem 3oalem1
StepHypRef Expression
1 3oalem1.1 . . . . 5 𝐵C
21cheli 28217 . . . 4 (𝑥𝐵𝑥 ∈ ℋ)
3 3oalem1.3 . . . . 5 𝑅C
43cheli 28217 . . . 4 (𝑦𝑅𝑦 ∈ ℋ)
52, 4anim12i 589 . . 3 ((𝑥𝐵𝑦𝑅) → (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ))
6 hvaddcl 27997 . . . . 5 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) ∈ ℋ)
7 eleq1 2718 . . . . 5 (𝑣 = (𝑥 + 𝑦) → (𝑣 ∈ ℋ ↔ (𝑥 + 𝑦) ∈ ℋ))
86, 7syl5ibrcom 237 . . . 4 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑣 = (𝑥 + 𝑦) → 𝑣 ∈ ℋ))
98imdistani 726 . . 3 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 = (𝑥 + 𝑦)) → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ))
105, 9sylan 487 . 2 (((𝑥𝐵𝑦𝑅) ∧ 𝑣 = (𝑥 + 𝑦)) → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ))
11 3oalem1.2 . . . . 5 𝐶C
1211cheli 28217 . . . 4 (𝑧𝐶𝑧 ∈ ℋ)
13 3oalem1.4 . . . . 5 𝑆C
1413cheli 28217 . . . 4 (𝑤𝑆𝑤 ∈ ℋ)
1512, 14anim12i 589 . . 3 ((𝑧𝐶𝑤𝑆) → (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ))
1615adantr 480 . 2 (((𝑧𝐶𝑤𝑆) ∧ 𝑣 = (𝑧 + 𝑤)) → (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ))
1710, 16anim12i 589 1 ((((𝑥𝐵𝑦𝑅) ∧ 𝑣 = (𝑥 + 𝑦)) ∧ ((𝑧𝐶𝑤𝑆) ∧ 𝑣 = (𝑧 + 𝑤))) → (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ) ∧ (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  (class class class)co 6690  chil 27904   + cva 27905   C cch 27914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-hilex 27984  ax-hfvadd 27985
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-sh 28192  df-ch 28206
This theorem is referenced by:  3oalem2  28650
  Copyright terms: Public domain W3C validator