Theorem tglinecom 28721

Description: Commutativity law for lines. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.)

Hypotheses

Ref	Expression
tglineelsb2.p	⊢ 𝐵 = (Base‘𝐺)
tglineelsb2.i	⊢ 𝐼 = (Itv‘𝐺)
tglineelsb2.l	⊢ 𝐿 = (LineG‘𝐺)
tglineelsb2.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglineelsb2.1	⊢ (𝜑 → 𝑃 ∈ 𝐵)
tglineelsb2.2	⊢ (𝜑 → 𝑄 ∈ 𝐵)
tglineelsb2.4	⊢ (𝜑 → 𝑃 ≠ 𝑄)

Assertion

Ref	Expression
tglinecom	⊢ (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃))

Proof of Theorem tglinecom

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tglineelsb2.p	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2		tglineelsb2.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
3		tglineelsb2.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
4		tglineelsb2.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝐺 ∈ TarskiG)
6		tglineelsb2.2	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄 ∈ 𝐵)
8		tglineelsb2.1	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
9	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑃 ∈ 𝐵)
10		tglineelsb2.4	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
11	1, 3, 2, 4, 8, 6, 10	tglnssp 28638	. . . . 5 ⊢ (𝜑 → (𝑃𝐿𝑄) ⊆ 𝐵)
12	11	sselda 3915	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ 𝐵)
13	10	necomd 2989	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑃)
14	13	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄 ≠ 𝑃)
15		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑃𝐿𝑄))
16	1, 2, 3, 5, 7, 9, 12, 14, 15	lncom 28708	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑄𝐿𝑃))
17	4	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝐺 ∈ TarskiG)
18	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃 ∈ 𝐵)
19	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑄 ∈ 𝐵)
20	1, 3, 2, 4, 6, 8, 13	tglnssp 28638	. . . . 5 ⊢ (𝜑 → (𝑄𝐿𝑃) ⊆ 𝐵)
21	20	sselda 3915	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ 𝐵)
22	10	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃 ≠ 𝑄)
23		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑄𝐿𝑃))
24	1, 2, 3, 17, 18, 19, 21, 22, 23	lncom 28708	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑃𝐿𝑄))
25	16, 24	impbida 806	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝑃𝐿𝑄) ↔ 𝑥 ∈ (𝑄𝐿𝑃)))
26	25	eqrdv 2737	1 ⊢ (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  TarskiGcstrkg 28513  Itvcitv 28519  LineGclng 28520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-trkgc 28534  df-trkgb 28535  df-trkgcb 28536  df-trkg 28539

This theorem is referenced by:  tglinethru  28722  coltr3  28734  footeq  28810  colperpexlem3  28818  mideulem2  28820  opphllem  28821  midex  28823  opphllem3  28835  opphllem5  28837  lmicom  28874  lmiisolem  28882  lnperpex  28889  trgcopy  28890