Theorem isperp 28805

Description: Property for 2 lines A, B to be perpendicular. Item (ii) of definition 8.11 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 16-Oct-2019.)

Hypotheses

Ref	Expression
isperp.p	⊢ 𝑃 = (Base‘𝐺)
isperp.d	⊢ − = (dist‘𝐺)
isperp.i	⊢ 𝐼 = (Itv‘𝐺)
isperp.l	⊢ 𝐿 = (LineG‘𝐺)
isperp.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
isperp.a	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
isperp.b	⊢ (𝜑 → 𝐵 ∈ ran 𝐿)

Assertion

Ref	Expression
isperp	⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))

Distinct variable groups:   𝑣,𝑢,𝑥,𝐴   𝑢,𝐵,𝑣,𝑥   𝑢,𝐺,𝑣,𝑥   𝜑,𝑢,𝑣,𝑥

Allowed substitution hints:   𝑃(𝑥,𝑣,𝑢)   𝐼(𝑥,𝑣,𝑢)   𝐿(𝑥,𝑣,𝑢)   − (𝑥,𝑣,𝑢)

Proof of Theorem isperp

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

Step	Hyp	Ref	Expression
1		df-br 5080	. . 3 ⊢ (𝐴(⟂G‘𝐺)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (⟂G‘𝐺))
2		df-perpg 28789	. . . . 5 ⊢ ⟂G = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))})
3		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
4	3	fveq2d 6838	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (LineG‘𝑔) = (LineG‘𝐺))
5		isperp.l	. . . . . . . . . . 11 ⊢ 𝐿 = (LineG‘𝐺)
6	4, 5	eqtr4di 2793	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (LineG‘𝑔) = 𝐿)
7	6	rneqd 5887	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ran (LineG‘𝑔) = ran 𝐿)
8	7	eleq2d 2826	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝑎 ∈ ran (LineG‘𝑔) ↔ 𝑎 ∈ ran 𝐿))
9	7	eleq2d 2826	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝑏 ∈ ran (LineG‘𝑔) ↔ 𝑏 ∈ ran 𝐿))
10	8, 9	anbi12d 638	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ↔ (𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿)))
11	3	fveq2d 6838	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (∟G‘𝑔) = (∟G‘𝐺))
12	11	eleq2d 2826	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
13	12	ralbidv 3163	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
14	13	rexralbidv 3206	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
15	10, 14	anbi12d 638	. . . . . 6 ⊢ ((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))))
16	15	opabbidv 5145	. . . . 5 ⊢ ((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))})
17		isperp.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
18	17	elexd 3456	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ V)
19	5	fvexi 6848	. . . . . . . 8 ⊢ 𝐿 ∈ V
20		rnexg 7849	. . . . . . . 8 ⊢ (𝐿 ∈ V → ran 𝐿 ∈ V)
21	19, 20	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ran 𝐿 ∈ V)
22	21, 21	xpexd 7701	. . . . . 6 ⊢ (𝜑 → (ran 𝐿 × ran 𝐿) ∈ V)
23		opabssxp 5717	. . . . . . 7 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ⊆ (ran 𝐿 × ran 𝐿)
24	23	a1i 11	. . . . . 6 ⊢ (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ⊆ (ran 𝐿 × ran 𝐿))
25	22, 24	ssexd 5259	. . . . 5 ⊢ (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ∈ V)
26	2, 16, 18, 25	fvmptd2 6951	. . . 4 ⊢ (𝜑 → (⟂G‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))})
27	26	eleq2d 2826	. . 3 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ (⟂G‘𝐺) ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))}))
28	1, 27	bitrid 284	. 2 ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))}))
29		isperp.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
30		isperp.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
31		ineq12 4151	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ∩ 𝑏) = (𝐴 ∩ 𝐵))
32		simpll 772	. . . . . 6 ⊢ (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ (𝑎 ∩ 𝑏)) → 𝑎 = 𝐴)
33		simpllr 781	. . . . . . 7 ⊢ ((((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ (𝑎 ∩ 𝑏)) ∧ 𝑢 ∈ 𝑎) → 𝑏 = 𝐵)
34	33	raleqdv 3298	. . . . . 6 ⊢ ((((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ (𝑎 ∩ 𝑏)) ∧ 𝑢 ∈ 𝑎) → (∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
35	32, 34	raleqbidva 3304	. . . . 5 ⊢ (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ (𝑎 ∩ 𝑏)) → (∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
36	31, 35	rexeqbidva 3305	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
37	36	opelopab2a 5484	. . 3 ⊢ ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
38	29, 30, 37	syl2anc 590	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
39	28, 38	bitrd 280	1 ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  Vcvv 3432   ∩ cin 3889   ⊆ wss 3890  ⟨cop 4568   class class class wbr 5079  {copab 5141   × cxp 5623  ran crn 5626  ‘cfv 6492  ⟨“cs3 14802  Basecbs 17177  distcds 17227  TarskiGcstrkg 28520  Itvcitv 28526  LineGclng 28527  ∟Gcrag 28786  ⟂Gcperpg 28788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-perpg 28789

This theorem is referenced by:  perpcom  28806  perpneq  28807  isperp2  28808