Theorem 3oalem1 29445

Description: Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
3oalem1.1	⊢ 𝐵 ∈ Cℋ
3oalem1.2	⊢ 𝐶 ∈ Cℋ
3oalem1.3	⊢ 𝑅 ∈ Cℋ
3oalem1.4	⊢ 𝑆 ∈ Cℋ

Assertion

Ref	Expression
3oalem1	⊢ ((((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 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

Step	Hyp	Ref	Expression
1		3oalem1.1	. . . . 5 ⊢ 𝐵 ∈ Cℋ
2	1	cheli 29015	. . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ℋ)
3		3oalem1.3	. . . . 5 ⊢ 𝑅 ∈ Cℋ
4	3	cheli 29015	. . . 4 ⊢ (𝑦 ∈ 𝑅 → 𝑦 ∈ ℋ)
5	2, 4	anim12i 615	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) → (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ))
6		hvaddcl 28795	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) ∈ ℋ)
7		eleq1 2877	. . . . 5 ⊢ (𝑣 = (𝑥 +ℎ 𝑦) → (𝑣 ∈ ℋ ↔ (𝑥 +ℎ 𝑦) ∈ ℋ))
8	6, 7	syl5ibrcom 250	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑣 = (𝑥 +ℎ 𝑦) → 𝑣 ∈ ℋ))
9	8	imdistani 572	. . 3 ⊢ (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ))
10	5, 9	sylan 583	. 2 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ))
11		3oalem1.2	. . . . 5 ⊢ 𝐶 ∈ Cℋ
12	11	cheli 29015	. . . 4 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ ℋ)
13		3oalem1.4	. . . . 5 ⊢ 𝑆 ∈ Cℋ
14	13	cheli 29015	. . . 4 ⊢ (𝑤 ∈ 𝑆 → 𝑤 ∈ ℋ)
15	12, 14	anim12i 615	. . 3 ⊢ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) → (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ))
16	15	adantr 484	. 2 ⊢ (((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤)) → (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ))
17	10, 16	anim12i 615	1 ⊢ ((((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤))) → (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ) ∧ (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  (class class class)co 7135   ℋchba 28702   +_ℎ cva 28703   C_ℋ cch 28712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-hilex 28782  ax-hfvadd 28783

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-sh 28990  df-ch 29004

This theorem is referenced by:  3oalem2  29446