Theorem perpcom 29215

Description: The "perpendicular" relation is symmetric. 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 4109	. . . . 5 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
3	2	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴))
4		ralcom 3294	. . . . 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 2763	. . . . . . . 8 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
10		isperp.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
11	10	ad3antrrr 743	. . . . . . . 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 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝐴 ∈ ran 𝐿)
14		simplrr 790	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑢 ∈ 𝐴)
15	5, 8, 7, 11, 13, 14	tglnpt 29039	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑢 ∈ 𝑃)
16		simpllr 788	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ (𝐴 ∩ 𝐵))
17	16	elin1d 4104	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ 𝐴)
18	5, 8, 7, 11, 13, 17	tglnpt 29039	. . . . . . . 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 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝐵 ∈ ran 𝐿)
21		simplrl 789	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑣 ∈ 𝐵)
22	5, 8, 7, 11, 20, 21	tglnpt 29039	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑣 ∈ 𝑃)
23		simpr 490	. . . . . . . 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 29200	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺))
25	10	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝐺 ∈ TarskiG)
26	19	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝐵 ∈ ran 𝐿)
27		simplrl 789	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑣 ∈ 𝐵)
28	5, 8, 7, 25, 26, 27	tglnpt 29039	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑣 ∈ 𝑃)
29	12	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝐴 ∈ ran 𝐿)
30		simpllr 788	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ (𝐴 ∩ 𝐵))
31	30	elin1d 4104	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ 𝐴)
32	5, 8, 7, 25, 29, 31	tglnpt 29039	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ 𝑃)
33		simplrr 790	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑢 ∈ 𝐴)
34	5, 8, 7, 25, 29, 33	tglnpt 29039	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑢 ∈ 𝑃)
35		simpr 490	. . . . . . . 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 29200	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))
37	24, 36	impbida 813	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)))
38	37	2ralbidva 3228	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → (∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)))
39	4, 38	bitrid 286	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)))
40	3, 39	rexeqbidva 3330	. . 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 29214	. . 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 29214	. . 3 ⊢ (𝜑 → (𝐵(⟂G‘𝐺)𝐴 ↔ ∃𝑥 ∈ (𝐵 ∩ 𝐴)∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)))
43	40, 41, 42	3bitr4d 314	. 2 ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ 𝐵(⟂G‘𝐺)𝐴))
44	1, 43	mpbid 235	1 ⊢ (𝜑 → 𝐵(⟂G‘𝐺)𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ∩ cin 4092   class class class wbr 5106  ran crn 5656  ‘cfv 6555  ⟨“cs3 15018  Basecbs 17413  distcds 17463  TarskiGcstrkg 28916  Itvcitv 28922  LineGclng 28923  pInvGcmir 29151  ∟Gcrag 29195  ⟂Gcperpg 29197

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-oadd 8505  df-er 8742  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-dju 10006  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-n0 12631  df-xnn0 12704  df-z 12718  df-uz 12990  df-fz 13664  df-fzo 13813  df-hash 14499  df-word 14683  df-concat 14740  df-s1 14767  df-s2 15024  df-s3 15025  df-trkgc 28937  df-trkgb 28938  df-trkgcb 28939  df-trkg 28942  df-mir 29152  df-rag 29196  df-perpg 29198

This theorem is used by:  hlperpnel  29228  colperpexlem3  29235  mideulem2  29237  midex  29240  opphllem5  29254  opphllem6  29255  opphl  29257  lmieu  29316  lnperpex  29336  trgcopy  29338  prlngex  29456  prlngmid2  29466