Theorem perpln2 28972

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 28957	. . . . . 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 489	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
3	2	fveq2d 6887	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (LineG‘𝑔) = (LineG‘𝐺))
4		perpln.l	. . . . . . . . . . . 12 ⊢ 𝐿 = (LineG‘𝐺)
5	3, 4	eqtr4di 2816	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (LineG‘𝑔) = 𝐿)
6	5	rneqd 5930	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ran (LineG‘𝑔) = ran 𝐿)
7	6	eleq2d 2849	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝑎 ∈ ran (LineG‘𝑔) ↔ 𝑎 ∈ ran 𝐿))
8	6	eleq2d 2849	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝑏 ∈ ran (LineG‘𝑔) ↔ 𝑏 ∈ ran 𝐿))
9	7, 8	anbi12d 643	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ↔ (𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿)))
10	2	fveq2d 6887	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (∟G‘𝑔) = (∟G‘𝐺))
11	10	eleq2d 2849	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
12	11	ralbidv 3188	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
13	12	rexralbidv 3231	. . . . . . . 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 643	. . . . . . 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 5178	. . . . . 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 3478	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V)
18	4	fvexi 6897	. . . . . . . . 9 ⊢ 𝐿 ∈ V
19		rnexg 7900	. . . . . . . . 9 ⊢ (𝐿 ∈ V → ran 𝐿 ∈ V)
20	18, 19	mp1i 14	. . . . . . . 8 ⊢ (𝜑 → ran 𝐿 ∈ V)
21	20, 20	xpexd 7751	. . . . . . 7 ⊢ (𝜑 → (ran 𝐿 × ran 𝐿) ∈ V)
22		opabssxp 5755	. . . . . . . 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 5296	. . . . . 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 7000	. . . . 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 5930	. . . 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 5932	. . . 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 3982	. . 3 ⊢ (𝜑 → ran (⟂G‘𝐺) ⊆ ran (ran 𝐿 × ran 𝐿))
29		rnxpss 6172	. . 3 ⊢ ran (ran 𝐿 × ran 𝐿) ⊆ ran 𝐿
30	28, 29	sstrdi 3950	. 2 ⊢ (𝜑 → ran (⟂G‘𝐺) ⊆ ran 𝐿)
31		relopabv 5810	. . . . . 6 ⊢ Rel {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))}
32	25	releqd 5767	. . . . . 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 261	. . . . 5 ⊢ (𝜑 → Rel (⟂G‘𝐺))
34		perpln.2	. . . . 5 ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)
35		brrelex12 5715	. . . . 5 ⊢ ((Rel (⟂G‘𝐺) ∧ 𝐴(⟂G‘𝐺)𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
36	33, 34, 35	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
37	36	simpld 499	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
38	36	simprd 500	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
39		brelrng 5933	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴(⟂G‘𝐺)𝐵) → 𝐵 ∈ ran (⟂G‘𝐺))
40	37, 38, 34, 39	syl3anc 1398	. 2 ⊢ (𝜑 → 𝐵 ∈ ran (⟂G‘𝐺))
41	30, 40	sseldd 3939	1 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ∩ cin 3905   ⊆ wss 3906   class class class wbr 5110  {copab 5174   × cxp 5661  ran crn 5664  Rel wrel 5668  ‘cfv 6538  ⟨“cs3 14881  TarskiGcstrkg 28677  LineGclng 28684  ∟Gcrag 28954  ⟂Gcperpg 28956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fv 6546  df-perpg 28957

This theorem is referenced by:  perpin  28986  hlperpnel  28987  mideulem2  28996  opphllem  28997  opphllem3  29011  opphllem5  29013  opphllem6  29014  trgcopy  29096  perpprlng  29181  prlngex  29182