Theorem lines 46029

Description: The lines passing through two different points in a left module (or any extended structure having a base set, an addition, and a scalar multiplication). (Contributed by AV, 14-Jan-2023.)

Hypotheses

Ref	Expression
lines.b	⊢ 𝐵 = (Base‘𝑊)
lines.l	⊢ 𝐿 = (LineM‘𝑊)
lines.s	⊢ 𝑆 = (Scalar‘𝑊)
lines.k	⊢ 𝐾 = (Base‘𝑆)
lines.p	⊢ · = ( ·𝑠 ‘𝑊)
lines.a	⊢ + = (+g‘𝑊)
lines.m	⊢ − = (-g‘𝑆)
lines.1	⊢ 1 = (1r‘𝑆)

Assertion

Ref	Expression
lines	⊢ (𝑊 ∈ 𝑉 → 𝐿 = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))

Distinct variable groups:   𝐵,𝑝,𝑥,𝑦   𝑡,𝐾   𝑡,𝑆   𝑊,𝑝,𝑡,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑡)   + (𝑥,𝑦,𝑡,𝑝)   𝑆(𝑥,𝑦,𝑝)   · (𝑥,𝑦,𝑡,𝑝)   1 (𝑥,𝑦,𝑡,𝑝)   𝐾(𝑥,𝑦,𝑝)   𝐿(𝑥,𝑦,𝑡,𝑝)   − (𝑥,𝑦,𝑡,𝑝)   𝑉(𝑥,𝑦,𝑡,𝑝)

Proof of Theorem lines

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lines.l	. 2 ⊢ 𝐿 = (LineM‘𝑊)
2		df-line 46027	. . 3 ⊢ LineM = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ ((Base‘𝑤) ∖ {𝑥}) ↦ {𝑝 ∈ (Base‘𝑤) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝑤))𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)(𝑡( ·𝑠 ‘𝑤)𝑦))}))
3		lines.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
4		fveq2 6768	. . . . . . 7 ⊢ (𝑊 = 𝑤 → (Base‘𝑊) = (Base‘𝑤))
5	3, 4	eqtrid 2791	. . . . . 6 ⊢ (𝑊 = 𝑤 → 𝐵 = (Base‘𝑤))
6	5	difeq1d 4060	. . . . . 6 ⊢ (𝑊 = 𝑤 → (𝐵 ∖ {𝑥}) = ((Base‘𝑤) ∖ {𝑥}))
7		lines.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑆)
8		lines.s	. . . . . . . . . . 11 ⊢ 𝑆 = (Scalar‘𝑊)
9		fveq2 6768	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → (Scalar‘𝑊) = (Scalar‘𝑤))
10	8, 9	eqtrid 2791	. . . . . . . . . 10 ⊢ (𝑊 = 𝑤 → 𝑆 = (Scalar‘𝑤))
11	10	fveq2d 6772	. . . . . . . . 9 ⊢ (𝑊 = 𝑤 → (Base‘𝑆) = (Base‘(Scalar‘𝑤)))
12	7, 11	eqtrid 2791	. . . . . . . 8 ⊢ (𝑊 = 𝑤 → 𝐾 = (Base‘(Scalar‘𝑤)))
13		lines.a	. . . . . . . . . . 11 ⊢ + = (+g‘𝑊)
14		fveq2 6768	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → (+g‘𝑊) = (+g‘𝑤))
15	13, 14	eqtrid 2791	. . . . . . . . . 10 ⊢ (𝑊 = 𝑤 → + = (+g‘𝑤))
16		lines.p	. . . . . . . . . . . 12 ⊢ · = ( ·𝑠 ‘𝑊)
17		fveq2 6768	. . . . . . . . . . . 12 ⊢ (𝑊 = 𝑤 → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑤))
18	16, 17	eqtrid 2791	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → · = ( ·𝑠 ‘𝑤))
19		lines.m	. . . . . . . . . . . . . 14 ⊢ − = (-g‘𝑆)
20	8	fveq2i 6771	. . . . . . . . . . . . . 14 ⊢ (-g‘𝑆) = (-g‘(Scalar‘𝑊))
21	19, 20	eqtri 2767	. . . . . . . . . . . . 13 ⊢ − = (-g‘(Scalar‘𝑊))
22		2fveq3 6773	. . . . . . . . . . . . 13 ⊢ (𝑊 = 𝑤 → (-g‘(Scalar‘𝑊)) = (-g‘(Scalar‘𝑤)))
23	21, 22	eqtrid 2791	. . . . . . . . . . . 12 ⊢ (𝑊 = 𝑤 → − = (-g‘(Scalar‘𝑤)))
24		lines.1	. . . . . . . . . . . . . 14 ⊢ 1 = (1r‘𝑆)
25	8	fveq2i 6771	. . . . . . . . . . . . . 14 ⊢ (1r‘𝑆) = (1r‘(Scalar‘𝑊))
26	24, 25	eqtri 2767	. . . . . . . . . . . . 13 ⊢ 1 = (1r‘(Scalar‘𝑊))
27		2fveq3 6773	. . . . . . . . . . . . 13 ⊢ (𝑊 = 𝑤 → (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑤)))
28	26, 27	eqtrid 2791	. . . . . . . . . . . 12 ⊢ (𝑊 = 𝑤 → 1 = (1r‘(Scalar‘𝑤)))
29		eqidd 2740	. . . . . . . . . . . 12 ⊢ (𝑊 = 𝑤 → 𝑡 = 𝑡)
30	23, 28, 29	oveq123d 7289	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → ( 1 − 𝑡) = ((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡))
31		eqidd 2740	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → 𝑥 = 𝑥)
32	18, 30, 31	oveq123d 7289	. . . . . . . . . 10 ⊢ (𝑊 = 𝑤 → (( 1 − 𝑡) · 𝑥) = (((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠 ‘𝑤)𝑥))
33	18	oveqd 7285	. . . . . . . . . 10 ⊢ (𝑊 = 𝑤 → (𝑡 · 𝑦) = (𝑡( ·𝑠 ‘𝑤)𝑦))
34	15, 32, 33	oveq123d 7289	. . . . . . . . 9 ⊢ (𝑊 = 𝑤 → ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)) = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)(𝑡( ·𝑠 ‘𝑤)𝑦)))
35	34	eqeq2d 2750	. . . . . . . 8 ⊢ (𝑊 = 𝑤 → (𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ 𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)(𝑡( ·𝑠 ‘𝑤)𝑦))))
36	12, 35	rexeqbidv 3335	. . . . . . 7 ⊢ (𝑊 = 𝑤 → (∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ ∃𝑡 ∈ (Base‘(Scalar‘𝑤))𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)(𝑡( ·𝑠 ‘𝑤)𝑦))))
37	5, 36	rabeqbidv 3418	. . . . . 6 ⊢ (𝑊 = 𝑤 → {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))} = {𝑝 ∈ (Base‘𝑤) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝑤))𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)(𝑡( ·𝑠 ‘𝑤)𝑦))})
38	5, 6, 37	mpoeq123dv 7341	. . . . 5 ⊢ (𝑊 = 𝑤 → (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}) = (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ ((Base‘𝑤) ∖ {𝑥}) ↦ {𝑝 ∈ (Base‘𝑤) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝑤))𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)(𝑡( ·𝑠 ‘𝑤)𝑦))}))
39	38	eqcomd 2745	. . . 4 ⊢ (𝑊 = 𝑤 → (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ ((Base‘𝑤) ∖ {𝑥}) ↦ {𝑝 ∈ (Base‘𝑤) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝑤))𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)(𝑡( ·𝑠 ‘𝑤)𝑦))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
40	39	eqcoms 2747	. . 3 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ ((Base‘𝑤) ∖ {𝑥}) ↦ {𝑝 ∈ (Base‘𝑤) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝑤))𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)(𝑡( ·𝑠 ‘𝑤)𝑦))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
41		elex 3448	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
42	3	fvexi 6782	. . . . 5 ⊢ 𝐵 ∈ V
43	42	difexi 5255	. . . . 5 ⊢ (𝐵 ∖ {𝑥}) ∈ V
44	42, 43	mpoex 7906	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}) ∈ V
45	44	a1i 11	. . 3 ⊢ (𝑊 ∈ 𝑉 → (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}) ∈ V)
46	2, 40, 41, 45	fvmptd3 6892	. 2 ⊢ (𝑊 ∈ 𝑉 → (LineM‘𝑊) = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
47	1, 46	eqtrid 2791	1 ⊢ (𝑊 ∈ 𝑉 → 𝐿 = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2109  ∃wrex 3066  {crab 3069  Vcvv 3430   ∖ cdif 3888  {csn 4566  ‘cfv 6430  (class class class)co 7268   ∈ cmpo 7270  Basecbs 16893  +_gcplusg 16943  Scalarcsca 16946   ·_𝑠 cvsca 16947  -_gcsg 18560  1_rcur 19718  Line_Mcline 46025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-line 46027

This theorem is referenced by:  line  46030  rrxlines  46031