Theorem perpln2 29192

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 29177	. . . . . 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 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
3	2	fveq2d 6890	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (LineG‘𝑔) = (LineG‘𝐺))
4		perpln.l	. . . . . . . . . . . 12 ⊢ 𝐿 = (LineG‘𝐺)
5	3, 4	eqtr4di 2813	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (LineG‘𝑔) = 𝐿)
6	5	rneqd 5919	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ran (LineG‘𝑔) = ran 𝐿)
7	6	eleq2d 2846	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝑎 ∈ ran (LineG‘𝑔) ↔ 𝑎 ∈ ran 𝐿))
8	6	eleq2d 2846	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝑏 ∈ ran (LineG‘𝑔) ↔ 𝑏 ∈ ran 𝐿))
9	7, 8	anbi12d 644	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ↔ (𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿)))
10	2	fveq2d 6890	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (∟G‘𝑔) = (∟G‘𝐺))
11	10	eleq2d 2846	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
12	11	ralbidv 3185	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
13	12	rexralbidv 3228	. . . . . . . 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 644	. . . . . . 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 5170	. . . . . 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 3473	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V)
18	4	fvexi 6900	. . . . . . . . 9 ⊢ 𝐿 ∈ V
19		rnexg 7915	. . . . . . . . 9 ⊢ (𝐿 ∈ V → ran 𝐿 ∈ V)
20	18, 19	mp1i 14	. . . . . . . 8 ⊢ (𝜑 → ran 𝐿 ∈ V)
21	20, 20	xpexd 7766	. . . . . . 7 ⊢ (𝜑 → (ran 𝐿 × ran 𝐿) ∈ V)
22		opabssxp 5742	. . . . . . . 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 5285	. . . . . 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 7003	. . . . 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 5919	. . . 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 5921	. . . 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 3952	. . 3 ⊢ (𝜑 → ran (⟂G‘𝐺) ⊆ ran (ran 𝐿 × ran 𝐿))
29		rnxpss 6163	. . 3 ⊢ ran (ran 𝐿 × ran 𝐿) ⊆ ran 𝐿
30	28, 29	sstrdi 3920	. 2 ⊢ (𝜑 → ran (⟂G‘𝐺) ⊆ ran 𝐿)
31		relopabv 5798	. . . . . 6 ⊢ Rel {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))}
32	25	releqd 5754	. . . . . 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 5702	. . . . 5 ⊢ ((Rel (⟂G‘𝐺) ∧ 𝐴(⟂G‘𝐺)𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
36	33, 34, 35	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
37	36	simpld 500	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
38	36	simprd 501	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
39		brelrng 5922	. . 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 3909	1 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089   class class class wbr 5102  {copab 5166   × cxp 5648  ran crn 5651  Rel wrel 5655  ‘cfv 6539  ⟨“cs3 14998  TarskiGcstrkg 28895  LineGclng 28902  ∟Gcrag 29174  ⟂Gcperpg 29176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6541  df-fv 6547  df-perpg 29177

This theorem is used by:  perpin  29206  hlperpnel  29207  mideulem2  29216  opphllem  29217  opphllem3  29231  opphllem5  29233  opphllem6  29234  trgcopy  29317  perpprlng  29434  prlngex  29435