Theorem oppcom 28722

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 28717	. . . . . 6 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
9	1, 8	mpbid 232	. . . . 5 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
10	9	simpld 494	. . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷))
11	10	simprd 495	. . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷)
12	10	simpld 494	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
13	9	simprd 495	. . . 4 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))
14		opphl.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
15	14	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
16	6	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 ∈ 𝑃)
17		opphl.l	. . . . . . . . 9 ⊢ 𝐿 = (LineG‘𝐺)
18	14	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝐺 ∈ TarskiG)
19		opphl.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
20	19	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
21		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝑡 ∈ 𝐷)
22	2, 17, 4, 18, 20, 21	tglnpt 28527	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝑡 ∈ 𝑃)
23	22	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ 𝑃)
24	7	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐵 ∈ 𝑃)
25		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐵))
26	2, 3, 4, 15, 16, 23, 24, 25	tgbtwncom 28466	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐵𝐼𝐴))
27	14	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐺 ∈ TarskiG)
28	7	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐵 ∈ 𝑃)
29	22	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡 ∈ 𝑃)
30	6	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐴 ∈ 𝑃)
31		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡 ∈ (𝐵𝐼𝐴))
32	2, 3, 4, 27, 28, 29, 30, 31	tgbtwncom 28466	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡 ∈ (𝐴𝐼𝐵))
33	26, 32	impbida 800	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (𝑡 ∈ (𝐴𝐼𝐵) ↔ 𝑡 ∈ (𝐵𝐼𝐴)))
34	33	rexbidva 3154	. . . 4 ⊢ (𝜑 → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐵𝐼𝐴)))
35	13, 34	mpbid 232	. . 3 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐵𝐼𝐴))
36	11, 12, 35	jca31 514	. 2 ⊢ (𝜑 → ((¬ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐵𝐼𝐴)))
37	2, 3, 4, 5, 7, 6	islnopp 28717	. 2 ⊢ (𝜑 → (𝐵𝑂𝐴 ↔ ((¬ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐵𝐼𝐴))))
38	36, 37	mpbird 257	1 ⊢ (𝜑 → 𝐵𝑂𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056   ∖ cdif 3894   class class class wbr 5089  {copab 5151  ran crn 5615  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  distcds 17170  TarskiGcstrkg 28405  Itvcitv 28411  LineGclng 28412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-cnv 5622  df-dm 5624  df-rn 5625  df-iota 6437  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-trkgc 28426  df-trkgb 28427  df-trkgcb 28428  df-trkg 28431

This theorem is referenced by:  opphllem2  28726  opphllem4  28728  opphllem5  28729  opphllem6  28730  lnperpex  28781