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Theorem 3oalem1 30653
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 30223 . . . 4 (𝑥𝐵𝑥 ∈ ℋ)
3 3oalem1.3 . . . . 5 𝑅C
43cheli 30223 . . . 4 (𝑦𝑅𝑦 ∈ ℋ)
52, 4anim12i 614 . . 3 ((𝑥𝐵𝑦𝑅) → (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ))
6 hvaddcl 30003 . . . . 5 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) ∈ ℋ)
7 eleq1 2822 . . . . 5 (𝑣 = (𝑥 + 𝑦) → (𝑣 ∈ ℋ ↔ (𝑥 + 𝑦) ∈ ℋ))
86, 7syl5ibrcom 247 . . . 4 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑣 = (𝑥 + 𝑦) → 𝑣 ∈ ℋ))
98imdistani 570 . . 3 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 = (𝑥 + 𝑦)) → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ))
105, 9sylan 581 . 2 (((𝑥𝐵𝑦𝑅) ∧ 𝑣 = (𝑥 + 𝑦)) → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ))
11 3oalem1.2 . . . . 5 𝐶C
1211cheli 30223 . . . 4 (𝑧𝐶𝑧 ∈ ℋ)
13 3oalem1.4 . . . . 5 𝑆C
1413cheli 30223 . . . 4 (𝑤𝑆𝑤 ∈ ℋ)
1512, 14anim12i 614 . . 3 ((𝑧𝐶𝑤𝑆) → (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ))
1615adantr 482 . 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 397   = wceq 1542  wcel 2107  (class class class)co 7361  chba 29910   + cva 29911   C cch 29920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-hilex 29990  ax-hfvadd 29991
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-sh 30198  df-ch 30212
This theorem is referenced by:  3oalem2  30654
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