Theorem lines 43467

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 43465	. . 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 6446	. . . . . . 7 ⊢ (𝑊 = 𝑤 → (Base‘𝑊) = (Base‘𝑤))
5	3, 4	syl5eq 2826	. . . . . 6 ⊢ (𝑊 = 𝑤 → 𝐵 = (Base‘𝑤))
6	5	difeq1d 3950	. . . . . 6 ⊢ (𝑊 = 𝑤 → (𝐵 ∖ {𝑥}) = ((Base‘𝑤) ∖ {𝑥}))
7		lines.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑆)
8		lines.s	. . . . . . . . . . 11 ⊢ 𝑆 = (Scalar‘𝑊)
9		fveq2 6446	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → (Scalar‘𝑊) = (Scalar‘𝑤))
10	8, 9	syl5eq 2826	. . . . . . . . . 10 ⊢ (𝑊 = 𝑤 → 𝑆 = (Scalar‘𝑤))
11	10	fveq2d 6450	. . . . . . . . 9 ⊢ (𝑊 = 𝑤 → (Base‘𝑆) = (Base‘(Scalar‘𝑤)))
12	7, 11	syl5eq 2826	. . . . . . . 8 ⊢ (𝑊 = 𝑤 → 𝐾 = (Base‘(Scalar‘𝑤)))
13		lines.a	. . . . . . . . . . 11 ⊢ + = (+g‘𝑊)
14		fveq2 6446	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → (+g‘𝑊) = (+g‘𝑤))
15	13, 14	syl5eq 2826	. . . . . . . . . 10 ⊢ (𝑊 = 𝑤 → + = (+g‘𝑤))
16		lines.p	. . . . . . . . . . . 12 ⊢ · = ( ·𝑠 ‘𝑊)
17		fveq2 6446	. . . . . . . . . . . 12 ⊢ (𝑊 = 𝑤 → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑤))
18	16, 17	syl5eq 2826	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → · = ( ·𝑠 ‘𝑤))
19		lines.m	. . . . . . . . . . . . . 14 ⊢ − = (-g‘𝑆)
20	8	fveq2i 6449	. . . . . . . . . . . . . 14 ⊢ (-g‘𝑆) = (-g‘(Scalar‘𝑊))
21	19, 20	eqtri 2802	. . . . . . . . . . . . 13 ⊢ − = (-g‘(Scalar‘𝑊))
22		2fveq3 6451	. . . . . . . . . . . . 13 ⊢ (𝑊 = 𝑤 → (-g‘(Scalar‘𝑊)) = (-g‘(Scalar‘𝑤)))
23	21, 22	syl5eq 2826	. . . . . . . . . . . 12 ⊢ (𝑊 = 𝑤 → − = (-g‘(Scalar‘𝑤)))
24		lines.1	. . . . . . . . . . . . . 14 ⊢ 1 = (1r‘𝑆)
25	8	fveq2i 6449	. . . . . . . . . . . . . 14 ⊢ (1r‘𝑆) = (1r‘(Scalar‘𝑊))
26	24, 25	eqtri 2802	. . . . . . . . . . . . 13 ⊢ 1 = (1r‘(Scalar‘𝑊))
27		2fveq3 6451	. . . . . . . . . . . . 13 ⊢ (𝑊 = 𝑤 → (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑤)))
28	26, 27	syl5eq 2826	. . . . . . . . . . . 12 ⊢ (𝑊 = 𝑤 → 1 = (1r‘(Scalar‘𝑤)))
29		eqidd 2779	. . . . . . . . . . . 12 ⊢ (𝑊 = 𝑤 → 𝑡 = 𝑡)
30	23, 28, 29	oveq123d 6943	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → ( 1 − 𝑡) = ((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡))
31		eqidd 2779	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → 𝑥 = 𝑥)
32	18, 30, 31	oveq123d 6943	. . . . . . . . . 10 ⊢ (𝑊 = 𝑤 → (( 1 − 𝑡) · 𝑥) = (((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠 ‘𝑤)𝑥))
33	18	oveqd 6939	. . . . . . . . . 10 ⊢ (𝑊 = 𝑤 → (𝑡 · 𝑦) = (𝑡( ·𝑠 ‘𝑤)𝑦))
34	15, 32, 33	oveq123d 6943	. . . . . . . . 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 2788	. . . . . . . 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 3327	. . . . . . 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 3392	. . . . . 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	mpt2eq123dv 6994	. . . . 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 2784	. . . 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 2786	. . 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 3414	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
42	3	fvexi 6460	. . . . 5 ⊢ 𝐵 ∈ V
43	42	difexi 5046	. . . . 5 ⊢ (𝐵 ∖ {𝑥}) ∈ V
44	42, 43	mpt2ex 7527	. . . 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 6564	. 2 ⊢ (𝑊 ∈ 𝑉 → (LineM‘𝑊) = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
47	1, 46	syl5eq 2826	1 ⊢ (𝑊 ∈ 𝑉 → 𝐿 = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))

Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2107  ∃wrex 3091  {crab 3094  Vcvv 3398   ∖ cdif 3789  {csn 4398  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924  Basecbs 16255  +_gcplusg 16338  Scalarcsca 16341   ·_𝑠 cvsca 16342  -_gcsg 17811  1_rcur 18888  Line_Mcline 43463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-line 43465

This theorem is referenced by:  line  43468  rrxlines  43469