Theorem perpcom 28721

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 4209	. . . . 5 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
3	2	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴))
4		ralcom 3289	. . . . 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 730	. . . . . . . 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 730	. . . . . . . . 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 28557	. . . . . . . 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 4204	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ 𝐴)
18	5, 8, 7, 11, 13, 17	tglnpt 28557	. . . . . . . 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 730	. . . . . . . . 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 28557	. . . . . . . 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 28706	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺))
25	10	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝐺 ∈ TarskiG)
26	19	ad3antrrr 730	. . . . . . . . 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 28557	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑣 ∈ 𝑃)
29	12	ad3antrrr 730	. . . . . . . . 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 4204	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴)) ∧ ⟨“𝑣𝑥𝑢”⟩ ∈ (∟G‘𝐺)) → 𝑥 ∈ 𝐴)
32	5, 8, 7, 25, 29, 31	tglnpt 28557	. . . . . . . 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 28557	. . . . . . . 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 28706	. . . . . . 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 3219	. . . . 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 3333	. . 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 28720	. . 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 28720	. . 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 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ∩ cin 3950   class class class wbr 5143  ran crn 5686  ‘cfv 6561  ⟨“cs3 14881  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  Itvcitv 28441  LineGclng 28442  pInvGcmir 28660  ∟Gcrag 28701  ⟂Gcperpg 28703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-s2 14887  df-s3 14888  df-trkgc 28456  df-trkgb 28457  df-trkgcb 28458  df-trkg 28461  df-mir 28661  df-rag 28702  df-perpg 28704

This theorem is referenced by:  hlperpnel  28733  colperpexlem3  28740  mideulem2  28742  midex  28745  opphllem5  28759  opphllem6  28760  opphl  28762  lmieu  28792  lnperpex  28811  trgcopy  28812