Theorem oppcom 27105

Description: Commutativity rule for "opposite" Theorem 9.2 of [Schwabhauser] p. 67. (Contributed by Thierry Arnoux, 19-Dec-2019.)

Hypotheses

Ref	Expression
hpg.p	⊢ 𝑃 = (Base‘𝐺)
hpg.d	⊢ − = (dist‘𝐺)
hpg.i	⊢ 𝐼 = (Itv‘𝐺)
hpg.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
opphl.l	⊢ 𝐿 = (LineG‘𝐺)
opphl.d	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
opphl.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
oppcom.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
oppcom.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
oppcom.o	⊢ (𝜑 → 𝐴𝑂𝐵)

Assertion

Ref	Expression
oppcom	⊢ (𝜑 → 𝐵𝑂𝐴)

Distinct variable groups:   𝐷,𝑎,𝑏   𝐼,𝑎,𝑏   𝑃,𝑎,𝑏   𝑡,𝐴   𝑡,𝐵   𝑡,𝐷   𝑡,𝐺   𝑡,𝐿   𝑡,𝐼   𝑡,𝑂   𝑡,𝑃   𝜑,𝑡   𝑡, −   𝑡,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐺(𝑎,𝑏)   𝐿(𝑎,𝑏)   − (𝑎,𝑏)   𝑂(𝑎,𝑏)

Proof of Theorem oppcom

Step	Hyp	Ref	Expression
1		oppcom.o	. . . . . 6 ⊢ (𝜑 → 𝐴𝑂𝐵)
2		hpg.p	. . . . . . 7 ⊢ 𝑃 = (Base‘𝐺)
3		hpg.d	. . . . . . 7 ⊢ − = (dist‘𝐺)
4		hpg.i	. . . . . . 7 ⊢ 𝐼 = (Itv‘𝐺)
5		hpg.o	. . . . . . 7 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
6		oppcom.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7		oppcom.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
8	2, 3, 4, 5, 6, 7	islnopp 27100	. . . . . 6 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
9	1, 8	mpbid 231	. . . . 5 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
10	9	simpld 495	. . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷))
11	10	simprd 496	. . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷)
12	10	simpld 495	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
13	9	simprd 496	. . . 4 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))
14		opphl.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
15	14	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
16	6	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 ∈ 𝑃)
17		opphl.l	. . . . . . . . 9 ⊢ 𝐿 = (LineG‘𝐺)
18	14	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝐺 ∈ TarskiG)
19		opphl.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
20	19	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
21		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝑡 ∈ 𝐷)
22	2, 17, 4, 18, 20, 21	tglnpt 26910	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝑡 ∈ 𝑃)
23	22	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ 𝑃)
24	7	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐵 ∈ 𝑃)
25		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐵))
26	2, 3, 4, 15, 16, 23, 24, 25	tgbtwncom 26849	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐵𝐼𝐴))
27	14	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐺 ∈ TarskiG)
28	7	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐵 ∈ 𝑃)
29	22	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡 ∈ 𝑃)
30	6	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐴 ∈ 𝑃)
31		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡 ∈ (𝐵𝐼𝐴))
32	2, 3, 4, 27, 28, 29, 30, 31	tgbtwncom 26849	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡 ∈ (𝐴𝐼𝐵))
33	26, 32	impbida 798	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (𝑡 ∈ (𝐴𝐼𝐵) ↔ 𝑡 ∈ (𝐵𝐼𝐴)))
34	33	rexbidva 3225	. . . 4 ⊢ (𝜑 → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐵𝐼𝐴)))
35	13, 34	mpbid 231	. . 3 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐵𝐼𝐴))
36	11, 12, 35	jca31 515	. 2 ⊢ (𝜑 → ((¬ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐵𝐼𝐴)))
37	2, 3, 4, 5, 7, 6	islnopp 27100	. 2 ⊢ (𝜑 → (𝐵𝑂𝐴 ↔ ((¬ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐵𝐼𝐴))))
38	36, 37	mpbird 256	1 ⊢ (𝜑 → 𝐵𝑂𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   ∖ cdif 3884   class class class wbr 5074  {copab 5136  ran crn 5590  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  distcds 16971  TarskiGcstrkg 26788  Itvcitv 26794  LineGclng 26795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-cnv 5597  df-dm 5599  df-rn 5600  df-iota 6391  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-trkgc 26809  df-trkgb 26810  df-trkgcb 26811  df-trkg 26814

This theorem is referenced by:  opphllem2  27109  opphllem4  27111  opphllem5  27112  opphllem6  27113  lnperpex  27164