Theorem perpcom 26978

Description: The "perpendicular" relation commutes. Theorem 8.12 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)
perpcom.1	⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)

Assertion

Ref	Expression
perpcom	⊢ (𝜑 → 𝐵(⟂G‘𝐺)𝐴)

Proof of Theorem perpcom

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

Step	Hyp	Ref	Expression
1		perpcom.1	. 2 ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)
2		incom 4131	. . . . 5 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
3	2	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴))
4		ralcom 3280	. . . . 5 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))
5		isperp.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘𝐺)
6		isperp.d	. . . . . . . 8 ⊢ − = (dist‘𝐺)
7		isperp.i	. . . . . . . 8 ⊢ 𝐼 = (Itv‘𝐺)
8		isperp.l	. . . . . . . 8 ⊢ 𝐿 = (LineG‘𝐺)
9		eqid 2738	. . . . . . . 8 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
10		isperp.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
11	10	ad3antrrr 726	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝐺 ∈ TarskiG)
12		isperp.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
13	12	ad3antrrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝐴 ∈ ran 𝐿)
14		simplrr 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑢 ∈ 𝐴)
15	5, 8, 7, 11, 13, 14	tglnpt 26814	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑢 ∈ 𝑃)
16		simpllr 772	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ (𝐴 ∩ 𝐵))
17	16	elin1d 4128	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ 𝐴)
18	5, 8, 7, 11, 13, 17	tglnpt 26814	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ 𝑃)
19		isperp.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
20	19	ad3antrrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝐵 ∈ ran 𝐿)
21		simplrl 773	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑣 ∈ 𝐵)
22	5, 8, 7, 11, 20, 21	tglnpt 26814	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑣 ∈ 𝑃)
23		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))
24	5, 6, 7, 8, 9, 11, 15, 18, 22, 23	ragcom 26963	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺))
25	10	ad3antrrr 726	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝐺 ∈ TarskiG)
26	19	ad3antrrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝐵 ∈ ran 𝐿)
27		simplrl 773	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑣 ∈ 𝐵)
28	5, 8, 7, 25, 26, 27	tglnpt 26814	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑣 ∈ 𝑃)
29	12	ad3antrrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝐴 ∈ ran 𝐿)
30		simpllr 772	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ (𝐴 ∩ 𝐵))
31	30	elin1d 4128	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ 𝐴)
32	5, 8, 7, 25, 29, 31	tglnpt 26814	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ 𝑃)
33		simplrr 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑢 ∈ 𝐴)
34	5, 8, 7, 25, 29, 33	tglnpt 26814	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑢 ∈ 𝑃)
35		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺))
36	5, 6, 7, 8, 9, 25, 28, 32, 34, 35	ragcom 26963	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))
37	24, 36	impbida 797	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)))
38	37	2ralbidva 3121	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → (∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)))
39	4, 38	syl5bb 282	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)))
40	3, 39	rexeqbidva 3346	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ∃𝑥 ∈ (𝐵 ∩ 𝐴)∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)))
41	5, 6, 7, 8, 10, 12, 19	isperp 26977	. . 3 ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
42	5, 6, 7, 8, 10, 19, 12	isperp 26977	. . 3 ⊢ (𝜑 → (𝐵(⟂G‘𝐺)𝐴 ↔ ∃𝑥 ∈ (𝐵 ∩ 𝐴)∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)))
43	40, 41, 42	3bitr4d 310	. 2 ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ 𝐵(⟂G‘𝐺)𝐴))
44	1, 43	mpbid 231	1 ⊢ (𝜑 → 𝐵(⟂G‘𝐺)𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ∩ cin 3882   class class class wbr 5070  ran crn 5581  ‘cfv 6418  ⟨“cs3 14483  Basecbs 16840  distcds 16897  TarskiGcstrkg 26693  Itvcitv 26699  LineGclng 26700  pInvGcmir 26917  ∟Gcrag 26958  ⟂Gcperpg 26960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-hash 13973  df-word 14146  df-concat 14202  df-s1 14229  df-s2 14489  df-s3 14490  df-trkgc 26713  df-trkgb 26714  df-trkgcb 26715  df-trkg 26718  df-mir 26918  df-rag 26959  df-perpg 26961

This theorem is referenced by:  hlperpnel  26990  colperpexlem3  26997  mideulem2  26999  midex  27002  opphllem5  27016  opphllem6  27017  opphl  27019  lmieu  27049  lnperpex  27068  trgcopy  27069