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Theorem 3oalem1 29433
 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 29003 . . . 4 (𝑥𝐵𝑥 ∈ ℋ)
3 3oalem1.3 . . . . 5 𝑅C
43cheli 29003 . . . 4 (𝑦𝑅𝑦 ∈ ℋ)
52, 4anim12i 614 . . 3 ((𝑥𝐵𝑦𝑅) → (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ))
6 hvaddcl 28783 . . . . 5 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) ∈ ℋ)
7 eleq1 2900 . . . . 5 (𝑣 = (𝑥 + 𝑦) → (𝑣 ∈ ℋ ↔ (𝑥 + 𝑦) ∈ ℋ))
86, 7syl5ibrcom 249 . . . 4 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑣 = (𝑥 + 𝑦) → 𝑣 ∈ ℋ))
98imdistani 571 . . 3 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 = (𝑥 + 𝑦)) → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ))
105, 9sylan 582 . 2 (((𝑥𝐵𝑦𝑅) ∧ 𝑣 = (𝑥 + 𝑦)) → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ))
11 3oalem1.2 . . . . 5 𝐶C
1211cheli 29003 . . . 4 (𝑧𝐶𝑧 ∈ ℋ)
13 3oalem1.4 . . . . 5 𝑆C
1413cheli 29003 . . . 4 (𝑤𝑆𝑤 ∈ ℋ)
1512, 14anim12i 614 . . 3 ((𝑧𝐶𝑤𝑆) → (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ))
1615adantr 483 . 2 (((𝑧𝐶𝑤𝑆) ∧ 𝑣 = (𝑧 + 𝑤)) → (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ))
1710, 16anim12i 614 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 398   = wceq 1533   ∈ wcel 2110  (class class class)co 7150   ℋchba 28690   +ℎ cva 28691   Cℋ cch 28700 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-hilex 28770  ax-hfvadd 28771 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-ov 7153  df-sh 28978  df-ch 28992 This theorem is referenced by:  3oalem2  29434
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