Theorem perpcom 28781

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 4149	. . . . 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 2736	. . . . . . . 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 28617	. . . . . . . 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 4144	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ 𝐴)
18	5, 8, 7, 11, 13, 17	tglnpt 28617	. . . . . . . 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 28617	. . . . . . . 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 28766	. . . . . . 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 28617	. . . . . . . 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 4144	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ 𝐴)
32	5, 8, 7, 25, 29, 31	tglnpt 28617	. . . . . . . 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 28617	. . . . . . . 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 28766	. . . . . . 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 3302	. . 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 28780	. . 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 28780	. . 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 3051  ∃wrex 3061   ∩ cin 3888   class class class wbr 5085  ran crn 5632  ‘cfv 6498  ⟨“cs3 14804  Basecbs 17179  distcds 17229  TarskiGcstrkg 28495  Itvcitv 28501  LineGclng 28502  pInvGcmir 28720  ∟Gcrag 28761  ⟂Gcperpg 28763

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-fz 13462  df-fzo 13609  df-hash 14293  df-word 14476  df-concat 14533  df-s1 14559  df-s2 14810  df-s3 14811  df-trkgc 28516  df-trkgb 28517  df-trkgcb 28518  df-trkg 28521  df-mir 28721  df-rag 28762  df-perpg 28764

This theorem is referenced by:  hlperpnel  28793  colperpexlem3  28800  mideulem2  28802  midex  28805  opphllem5  28819  opphllem6  28820  opphl  28822  lmieu  28852  lnperpex  28871  trgcopy  28872