Theorem 3oalem1 31919

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 31489	. . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ℋ)
3		3oalem1.3	. . . . 5 ⊢ 𝑅 ∈ Cℋ
4	3	cheli 31489	. . . 4 ⊢ (𝑦 ∈ 𝑅 → 𝑦 ∈ ℋ)
5	2, 4	anim12i 624	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) → (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ))
6		hvaddcl 31269	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) ∈ ℋ)
7		eleq1 2853	. . . . 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 578	. . 3 ⊢ (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ))
10	5, 9	sylan 591	. 2 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ))
11		3oalem1.2	. . . . 5 ⊢ 𝐶 ∈ Cℋ
12	11	cheli 31489	. . . 4 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ ℋ)
13		3oalem1.4	. . . . 5 ⊢ 𝑆 ∈ Cℋ
14	13	cheli 31489	. . . 4 ⊢ (𝑤 ∈ 𝑆 → 𝑤 ∈ ℋ)
15	12, 14	anim12i 624	. . 3 ⊢ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) → (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ))
16	15	adantr 485	. 2 ⊢ (((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤)) → (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ))
17	10, 16	anim12i 624	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 400   = wceq 1563   ∈ wcel 2145  (class class class)co 7400   ℋchba 31176   +_ℎ cva 31177   C_ℋ cch 31186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pr 5394  ax-hilex 31256  ax-hfvadd 31257

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-ov 7403  df-sh 31464  df-ch 31478

This theorem is referenced by:  3oalem2  31920