Theorem lines 46937

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 46935	. . 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 6847	. . . . . . 7 ⊢ (𝑊 = 𝑤 → (Base‘𝑊) = (Base‘𝑤))
5	3, 4	eqtrid 2783	. . . . . 6 ⊢ (𝑊 = 𝑤 → 𝐵 = (Base‘𝑤))
6	5	difeq1d 4086	. . . . . 6 ⊢ (𝑊 = 𝑤 → (𝐵 ∖ {𝑥}) = ((Base‘𝑤) ∖ {𝑥}))
7		lines.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑆)
8		lines.s	. . . . . . . . . . 11 ⊢ 𝑆 = (Scalar‘𝑊)
9		fveq2 6847	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → (Scalar‘𝑊) = (Scalar‘𝑤))
10	8, 9	eqtrid 2783	. . . . . . . . . 10 ⊢ (𝑊 = 𝑤 → 𝑆 = (Scalar‘𝑤))
11	10	fveq2d 6851	. . . . . . . . 9 ⊢ (𝑊 = 𝑤 → (Base‘𝑆) = (Base‘(Scalar‘𝑤)))
12	7, 11	eqtrid 2783	. . . . . . . 8 ⊢ (𝑊 = 𝑤 → 𝐾 = (Base‘(Scalar‘𝑤)))
13		lines.a	. . . . . . . . . . 11 ⊢ + = (+g‘𝑊)
14		fveq2 6847	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → (+g‘𝑊) = (+g‘𝑤))
15	13, 14	eqtrid 2783	. . . . . . . . . 10 ⊢ (𝑊 = 𝑤 → + = (+g‘𝑤))
16		lines.p	. . . . . . . . . . . 12 ⊢ · = ( ·𝑠 ‘𝑊)
17		fveq2 6847	. . . . . . . . . . . 12 ⊢ (𝑊 = 𝑤 → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑤))
18	16, 17	eqtrid 2783	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → · = ( ·𝑠 ‘𝑤))
19		lines.m	. . . . . . . . . . . . . 14 ⊢ − = (-g‘𝑆)
20	8	fveq2i 6850	. . . . . . . . . . . . . 14 ⊢ (-g‘𝑆) = (-g‘(Scalar‘𝑊))
21	19, 20	eqtri 2759	. . . . . . . . . . . . 13 ⊢ − = (-g‘(Scalar‘𝑊))
22		2fveq3 6852	. . . . . . . . . . . . 13 ⊢ (𝑊 = 𝑤 → (-g‘(Scalar‘𝑊)) = (-g‘(Scalar‘𝑤)))
23	21, 22	eqtrid 2783	. . . . . . . . . . . 12 ⊢ (𝑊 = 𝑤 → − = (-g‘(Scalar‘𝑤)))
24		lines.1	. . . . . . . . . . . . . 14 ⊢ 1 = (1r‘𝑆)
25	8	fveq2i 6850	. . . . . . . . . . . . . 14 ⊢ (1r‘𝑆) = (1r‘(Scalar‘𝑊))
26	24, 25	eqtri 2759	. . . . . . . . . . . . 13 ⊢ 1 = (1r‘(Scalar‘𝑊))
27		2fveq3 6852	. . . . . . . . . . . . 13 ⊢ (𝑊 = 𝑤 → (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑤)))
28	26, 27	eqtrid 2783	. . . . . . . . . . . 12 ⊢ (𝑊 = 𝑤 → 1 = (1r‘(Scalar‘𝑤)))
29		eqidd 2732	. . . . . . . . . . . 12 ⊢ (𝑊 = 𝑤 → 𝑡 = 𝑡)
30	23, 28, 29	oveq123d 7383	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → ( 1 − 𝑡) = ((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡))
31		eqidd 2732	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → 𝑥 = 𝑥)
32	18, 30, 31	oveq123d 7383	. . . . . . . . . 10 ⊢ (𝑊 = 𝑤 → (( 1 − 𝑡) · 𝑥) = (((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠 ‘𝑤)𝑥))
33	18	oveqd 7379	. . . . . . . . . 10 ⊢ (𝑊 = 𝑤 → (𝑡 · 𝑦) = (𝑡( ·𝑠 ‘𝑤)𝑦))
34	15, 32, 33	oveq123d 7383	. . . . . . . . 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 2742	. . . . . . . 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 3318	. . . . . . 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 3422	. . . . . 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 7437	. . . . 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 2737	. . . 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 2739	. . 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 3464	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
42	3	fvexi 6861	. . . . 5 ⊢ 𝐵 ∈ V
43	42	difexi 5290	. . . . 5 ⊢ (𝐵 ∖ {𝑥}) ∈ V
44	42, 43	mpoex 8017	. . . 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 6976	. 2 ⊢ (𝑊 ∈ 𝑉 → (LineM‘𝑊) = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
47	1, 46	eqtrid 2783	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 2106  ∃wrex 3069  {crab 3405  Vcvv 3446   ∖ cdif 3910  {csn 4591  ‘cfv 6501  (class class class)co 7362   ∈ cmpo 7364  Basecbs 17094  +_gcplusg 17147  Scalarcsca 17150   ·_𝑠 cvsca 17151  -_gcsg 18764  1_rcur 19927  Line_Mcline 46933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-line 46935

This theorem is referenced by:  line  46938  rrxlines  46939