Theorem perpcom 28789

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 4162	. . . . 5 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
3	2	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴))
4		ralcom 3265	. . . . 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 2737	. . . . . . . 8 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
10		isperp.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
11	10	ad3antrrr 731	. . . . . . . 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 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝐴 ∈ ran 𝐿)
14		simplrr 778	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑢 ∈ 𝐴)
15	5, 8, 7, 11, 13, 14	tglnpt 28625	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑢 ∈ 𝑃)
16		simpllr 776	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ (𝐴 ∩ 𝐵))
17	16	elin1d 4157	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ 𝐴)
18	5, 8, 7, 11, 13, 17	tglnpt 28625	. . . . . . . 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 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝐵 ∈ ran 𝐿)
21		simplrl 777	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑣 ∈ 𝐵)
22	5, 8, 7, 11, 20, 21	tglnpt 28625	. . . . . . . 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 28774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺))
25	10	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝐺 ∈ TarskiG)
26	19	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝐵 ∈ ran 𝐿)
27		simplrl 777	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑣 ∈ 𝐵)
28	5, 8, 7, 25, 26, 27	tglnpt 28625	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑣 ∈ 𝑃)
29	12	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝐴 ∈ ran 𝐿)
30		simpllr 776	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ (𝐴 ∩ 𝐵))
31	30	elin1d 4157	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ 𝐴)
32	5, 8, 7, 25, 29, 31	tglnpt 28625	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ 𝑃)
33		simplrr 778	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑢 ∈ 𝐴)
34	5, 8, 7, 25, 29, 33	tglnpt 28625	. . . . . . . 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 28774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))
37	24, 36	impbida 801	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)))
38	37	2ralbidva 3199	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → (∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)))
39	4, 38	bitrid 283	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)))
40	3, 39	rexeqbidva 3304	. . 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 28788	. . 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 28788	. . 3 ⊢ (𝜑 → (𝐵(⟂G‘𝐺)𝐴 ↔ ∃𝑥 ∈ (𝐵 ∩ 𝐴)∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)))
43	40, 41, 42	3bitr4d 311	. 2 ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ 𝐵(⟂G‘𝐺)𝐴))
44	1, 43	mpbid 232	1 ⊢ (𝜑 → 𝐵(⟂G‘𝐺)𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061   ∩ cin 3901   class class class wbr 5099  ran crn 5626  ‘cfv 6493  ⟨“cs3 14769  Basecbs 17140  distcds 17190  TarskiGcstrkg 28503  Itvcitv 28509  LineGclng 28510  pInvGcmir 28728  ∟Gcrag 28769  ⟂Gcperpg 28771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-oadd 8403  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9817  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-3 12213  df-n0 12406  df-xnn0 12479  df-z 12493  df-uz 12756  df-fz 13428  df-fzo 13575  df-hash 14258  df-word 14441  df-concat 14498  df-s1 14524  df-s2 14775  df-s3 14776  df-trkgc 28524  df-trkgb 28525  df-trkgcb 28526  df-trkg 28529  df-mir 28729  df-rag 28770  df-perpg 28772

This theorem is referenced by:  hlperpnel  28801  colperpexlem3  28808  mideulem2  28810  midex  28813  opphllem5  28827  opphllem6  28828  opphl  28830  lmieu  28860  lnperpex  28879  trgcopy  28880