Theorem lines 49095

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 49093	. . 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 6842	. . . . . . 7 ⊢ (𝑊 = 𝑤 → (Base‘𝑊) = (Base‘𝑤))
5	3, 4	eqtrid 2784	. . . . . 6 ⊢ (𝑊 = 𝑤 → 𝐵 = (Base‘𝑤))
6	5	difeq1d 4079	. . . . . 6 ⊢ (𝑊 = 𝑤 → (𝐵 ∖ {𝑥}) = ((Base‘𝑤) ∖ {𝑥}))
7		lines.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑆)
8		lines.s	. . . . . . . . . . 11 ⊢ 𝑆 = (Scalar‘𝑊)
9		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → (Scalar‘𝑊) = (Scalar‘𝑤))
10	8, 9	eqtrid 2784	. . . . . . . . . 10 ⊢ (𝑊 = 𝑤 → 𝑆 = (Scalar‘𝑤))
11	10	fveq2d 6846	. . . . . . . . 9 ⊢ (𝑊 = 𝑤 → (Base‘𝑆) = (Base‘(Scalar‘𝑤)))
12	7, 11	eqtrid 2784	. . . . . . . 8 ⊢ (𝑊 = 𝑤 → 𝐾 = (Base‘(Scalar‘𝑤)))
13		lines.a	. . . . . . . . . . 11 ⊢ + = (+g‘𝑊)
14		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → (+g‘𝑊) = (+g‘𝑤))
15	13, 14	eqtrid 2784	. . . . . . . . . 10 ⊢ (𝑊 = 𝑤 → + = (+g‘𝑤))
16		lines.p	. . . . . . . . . . . 12 ⊢ · = ( ·𝑠 ‘𝑊)
17		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑊 = 𝑤 → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑤))
18	16, 17	eqtrid 2784	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → · = ( ·𝑠 ‘𝑤))
19		lines.m	. . . . . . . . . . . . . 14 ⊢ − = (-g‘𝑆)
20	8	fveq2i 6845	. . . . . . . . . . . . . 14 ⊢ (-g‘𝑆) = (-g‘(Scalar‘𝑊))
21	19, 20	eqtri 2760	. . . . . . . . . . . . 13 ⊢ − = (-g‘(Scalar‘𝑊))
22		2fveq3 6847	. . . . . . . . . . . . 13 ⊢ (𝑊 = 𝑤 → (-g‘(Scalar‘𝑊)) = (-g‘(Scalar‘𝑤)))
23	21, 22	eqtrid 2784	. . . . . . . . . . . 12 ⊢ (𝑊 = 𝑤 → − = (-g‘(Scalar‘𝑤)))
24		lines.1	. . . . . . . . . . . . . 14 ⊢ 1 = (1r‘𝑆)
25	8	fveq2i 6845	. . . . . . . . . . . . . 14 ⊢ (1r‘𝑆) = (1r‘(Scalar‘𝑊))
26	24, 25	eqtri 2760	. . . . . . . . . . . . 13 ⊢ 1 = (1r‘(Scalar‘𝑊))
27		2fveq3 6847	. . . . . . . . . . . . 13 ⊢ (𝑊 = 𝑤 → (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑤)))
28	26, 27	eqtrid 2784	. . . . . . . . . . . 12 ⊢ (𝑊 = 𝑤 → 1 = (1r‘(Scalar‘𝑤)))
29		eqidd 2738	. . . . . . . . . . . 12 ⊢ (𝑊 = 𝑤 → 𝑡 = 𝑡)
30	23, 28, 29	oveq123d 7389	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → ( 1 − 𝑡) = ((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡))
31		eqidd 2738	. . . . . . . . . . 11 ⊢ (𝑊 = 𝑤 → 𝑥 = 𝑥)
32	18, 30, 31	oveq123d 7389	. . . . . . . . . 10 ⊢ (𝑊 = 𝑤 → (( 1 − 𝑡) · 𝑥) = (((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠 ‘𝑤)𝑥))
33	18	oveqd 7385	. . . . . . . . . 10 ⊢ (𝑊 = 𝑤 → (𝑡 · 𝑦) = (𝑡( ·𝑠 ‘𝑤)𝑦))
34	15, 32, 33	oveq123d 7389	. . . . . . . . 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 2748	. . . . . . . 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 3319	. . . . . . 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 3419	. . . . . 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 7443	. . . . 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 2743	. . . 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 2745	. . 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 3463	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
42	3	fvexi 6856	. . . . 5 ⊢ 𝐵 ∈ V
43	42	difexi 5277	. . . . 5 ⊢ (𝐵 ∖ {𝑥}) ∈ V
44	42, 43	mpoex 8033	. . . 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 6973	. 2 ⊢ (𝑊 ∈ 𝑉 → (LineM‘𝑊) = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
47	1, 46	eqtrid 2784	1 ⊢ (𝑊 ∈ 𝑉 → 𝐿 = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  {crab 3401  Vcvv 3442   ∖ cdif 3900  {csn 4582  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  Basecbs 17148  +_gcplusg 17189  Scalarcsca 17192   ·_𝑠 cvsca 17193  -_gcsg 18880  1_rcur 20131  Line_Mcline 49091

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-line 49093

This theorem is referenced by:  line  49096  rrxlines  49097