Theorem perpln2 27653

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
perpln2	⊢ (𝜑 → 𝐵 ∈ ran 𝐿)

Proof of Theorem perpln2

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

Step	Hyp	Ref	Expression
1		df-perpg 27638	. . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
3	2	fveq2d 6846	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (LineG‘𝑔) = (LineG‘𝐺))
4		perpln.l	. . . . . . . . . . . 12 ⊢ 𝐿 = (LineG‘𝐺)
5	3, 4	eqtr4di 2794	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (LineG‘𝑔) = 𝐿)
6	5	rneqd 5893	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ran (LineG‘𝑔) = ran 𝐿)
7	6	eleq2d 2823	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝑎 ∈ ran (LineG‘𝑔) ↔ 𝑎 ∈ ran 𝐿))
8	6	eleq2d 2823	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝑏 ∈ ran (LineG‘𝑔) ↔ 𝑏 ∈ ran 𝐿))
9	7, 8	anbi12d 631	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ↔ (𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿)))
10	2	fveq2d 6846	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (∟G‘𝑔) = (∟G‘𝐺))
11	10	eleq2d 2823	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
12	11	ralbidv 3174	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
13	12	rexralbidv 3214	. . . . . . . 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 631	. . . . . . 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 5171	. . . . . 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 3465	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V)
18	4	fvexi 6856	. . . . . . . . 9 ⊢ 𝐿 ∈ V
19		rnexg 7841	. . . . . . . . 9 ⊢ (𝐿 ∈ V → ran 𝐿 ∈ V)
20	18, 19	mp1i 13	. . . . . . . 8 ⊢ (𝜑 → ran 𝐿 ∈ V)
21	20, 20	xpexd 7685	. . . . . . 7 ⊢ (𝜑 → (ran 𝐿 × ran 𝐿) ∈ V)
22		opabssxp 5724	. . . . . . . 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 5281	. . . . . 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 6956	. . . . 5 ⊢ (𝜑 → (⟂G‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))})
26	25	rneqd 5893	. . . 4 ⊢ (𝜑 → ran (⟂G‘𝐺) = ran {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))})
27	22	rnssi 5895	. . . 4 ⊢ ran {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ⊆ ran (ran 𝐿 × ran 𝐿)
28	26, 27	eqsstrdi 3998	. . 3 ⊢ (𝜑 → ran (⟂G‘𝐺) ⊆ ran (ran 𝐿 × ran 𝐿))
29		rnxpss 6124	. . 3 ⊢ ran (ran 𝐿 × ran 𝐿) ⊆ ran 𝐿
30	28, 29	sstrdi 3956	. 2 ⊢ (𝜑 → ran (⟂G‘𝐺) ⊆ ran 𝐿)
31		relopabv 5777	. . . . . 6 ⊢ Rel {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))}
32	25	releqd 5734	. . . . . 6 ⊢ (𝜑 → (Rel (⟂G‘𝐺) ↔ Rel {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))}))
33	31, 32	mpbiri 257	. . . . 5 ⊢ (𝜑 → Rel (⟂G‘𝐺))
34		perpln.2	. . . . 5 ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)
35		brrelex12 5684	. . . . 5 ⊢ ((Rel (⟂G‘𝐺) ∧ 𝐴(⟂G‘𝐺)𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
36	33, 34, 35	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
37	36	simpld 495	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
38	36	simprd 496	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
39		brelrng 5896	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴(⟂G‘𝐺)𝐵) → 𝐵 ∈ ran (⟂G‘𝐺))
40	37, 38, 34, 39	syl3anc 1371	. 2 ⊢ (𝜑 → 𝐵 ∈ ran (⟂G‘𝐺))
41	30, 40	sseldd 3945	1 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  Vcvv 3445   ∩ cin 3909   ⊆ wss 3910   class class class wbr 5105  {copab 5167   × cxp 5631  ran crn 5634  Rel wrel 5638  ‘cfv 6496  ⟨“cs3 14731  TarskiGcstrkg 27369  LineGclng 27376  ∟Gcrag 27635  ⟂Gcperpg 27637

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fv 6504  df-perpg 27638

This theorem is referenced by:  hlperpnel  27667  mideulem2  27676  opphllem  27677  opphllem3  27691  opphllem5  27693  opphllem6  27694  trgcopy  27746