Theorem perpln1 28644

Description: Derive a line from perpendicularity. (Contributed by Thierry Arnoux, 27-Nov-2019.)

Hypotheses

Ref	Expression
perpln.l	⊢ 𝐿 = (LineG‘𝐺)
perpln.1	⊢ (𝜑 → 𝐺 ∈ TarskiG)
perpln.2	⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)

Assertion

Ref	Expression
perpln1	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)

Proof of Theorem perpln1

Dummy variables 𝑎 𝑏 𝑔 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-perpg 28630	. . . . . 6 ⊢ ⟂G = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))})
2		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
3	2	fveq2d 6865	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (LineG‘𝑔) = (LineG‘𝐺))
4		perpln.l	. . . . . . . . . . . 12 ⊢ 𝐿 = (LineG‘𝐺)
5	3, 4	eqtr4di 2783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (LineG‘𝑔) = 𝐿)
6	5	rneqd 5905	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ran (LineG‘𝑔) = ran 𝐿)
7	6	eleq2d 2815	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝑎 ∈ ran (LineG‘𝑔) ↔ 𝑎 ∈ ran 𝐿))
8	6	eleq2d 2815	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝑏 ∈ ran (LineG‘𝑔) ↔ 𝑏 ∈ ran 𝐿))
9	7, 8	anbi12d 632	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ↔ (𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿)))
10	2	fveq2d 6865	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (∟G‘𝑔) = (∟G‘𝐺))
11	10	eleq2d 2815	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
12	11	ralbidv 3157	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
13	12	rexralbidv 3204	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
14	9, 13	anbi12d 632	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔)) ↔ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))))
15	14	opabbidv 5176	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))})
16		perpln.1	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
17	16	elexd 3474	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V)
18	4	fvexi 6875	. . . . . . . . 9 ⊢ 𝐿 ∈ V
19		rnexg 7881	. . . . . . . . 9 ⊢ (𝐿 ∈ V → ran 𝐿 ∈ V)
20	18, 19	mp1i 13	. . . . . . . 8 ⊢ (𝜑 → ran 𝐿 ∈ V)
21	20, 20	xpexd 7730	. . . . . . 7 ⊢ (𝜑 → (ran 𝐿 × ran 𝐿) ∈ V)
22		opabssxp 5734	. . . . . . . 8 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ⊆ (ran 𝐿 × ran 𝐿)
23	22	a1i 11	. . . . . . 7 ⊢ (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ⊆ (ran 𝐿 × ran 𝐿))
24	21, 23	ssexd 5282	. . . . . 6 ⊢ (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ∈ V)
25	1, 15, 17, 24	fvmptd2 6979	. . . . 5 ⊢ (𝜑 → (⟂G‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))})
26		anass 468	. . . . . 6 ⊢ (((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) ↔ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))))
27	26	opabbii 5177	. . . . 5 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))}
28	25, 27	eqtrdi 2781	. . . 4 ⊢ (𝜑 → (⟂G‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))})
29	28	dmeqd 5872	. . 3 ⊢ (𝜑 → dom (⟂G‘𝐺) = dom {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))})
30		dmopabss 5885	. . 3 ⊢ dom {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))} ⊆ ran 𝐿
31	29, 30	eqsstrdi 3994	. 2 ⊢ (𝜑 → dom (⟂G‘𝐺) ⊆ ran 𝐿)
32		relopabv 5787	. . . . . 6 ⊢ Rel {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))}
33	25	releqd 5744	. . . . . 6 ⊢ (𝜑 → (Rel (⟂G‘𝐺) ↔ Rel {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))}))
34	32, 33	mpbiri 258	. . . . 5 ⊢ (𝜑 → Rel (⟂G‘𝐺))
35		perpln.2	. . . . 5 ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)
36		brrelex12 5693	. . . . 5 ⊢ ((Rel (⟂G‘𝐺) ∧ 𝐴(⟂G‘𝐺)𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
37	34, 35, 36	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
38	37	simpld 494	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
39	37	simprd 495	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
40		breldmg 5876	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴(⟂G‘𝐺)𝐵) → 𝐴 ∈ dom (⟂G‘𝐺))
41	38, 39, 35, 40	syl3anc 1373	. 2 ⊢ (𝜑 → 𝐴 ∈ dom (⟂G‘𝐺))
42	31, 41	sseldd 3950	1 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ∩ cin 3916   ⊆ wss 3917   class class class wbr 5110  {copab 5172   × cxp 5639  dom cdm 5641  ran crn 5642  Rel wrel 5646  ‘cfv 6514  ⟨“cs3 14815  TarskiGcstrkg 28361  LineGclng 28368  ∟Gcrag 28627  ⟂Gcperpg 28629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fv 6522  df-perpg 28630

This theorem is referenced by:  footne  28657  footeq  28658  perpdragALT  28661  perpdrag  28662  colperp  28663  midex  28671  opphl  28688  lmieu  28718  lnperpex  28737  trgcopy  28738