rrxline - Mathbox for Alexander van der Vekens

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Theorem rrxline 49239

Description: The line passing through the two different points 𝑋 and 𝑌 in a generalized real Euclidean space of finite dimension. (Contributed by AV, 14-Jan-2023.)

**Hypotheses**
Ref	Expression
rrxlines.e	⊢ 𝐸 = (ℝ^‘𝐼)
rrxlines.p	⊢ 𝑃 = (ℝ ↑_m 𝐼)
rrxlines.l	⊢ 𝐿 = (Line_M‘𝐸)
rrxlines.m	⊢ · = ( ·_𝑠 ‘𝐸)
rrxlines.a	⊢ + = (+_g‘𝐸)

**Assertion**
Ref	Expression
rrxline	⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))})

Distinct variable groups: 𝐸,𝑝,𝑡 𝐼,𝑝,𝑡 𝑃,𝑝 𝑋,𝑝,𝑡 𝑌,𝑝,𝑡 Allowed substitution hints: 𝑃(𝑡) + (𝑡,𝑝) · (𝑡,𝑝) 𝐿(𝑡,𝑝)

Proof of Theorem rrxline Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrxlines.e	. . . . 5 ⊢ 𝐸 = (ℝ^‘𝐼)
2		rrxlines.p	. . . . 5 ⊢ 𝑃 = (ℝ ↑_m 𝐼)
3		rrxlines.l	. . . . 5 ⊢ 𝐿 = (Line_M‘𝐸)
4		rrxlines.m	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝐸)
5		rrxlines.a	. . . . 5 ⊢ + = (+_g‘𝐸)
6	1, 2, 3, 4, 5	rrxlines 49238	. . . 4 ⊢ (𝐼 ∈ Fin → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
7	6	oveqd 7377	. . 3 ⊢ (𝐼 ∈ Fin → (𝑋𝐿𝑌) = (𝑋(𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))})𝑌))
8	7	adantr 482	. 2 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → (𝑋𝐿𝑌) = (𝑋(𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))})𝑌))
9		eqidd 2742	. . 3 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
10		simpl 484	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
11	10	oveq2d 7376	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((1 − 𝑡) · 𝑥) = ((1 − 𝑡) · 𝑋))
12		simpr 486	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
13	12	oveq2d 7376	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑡 · 𝑦) = (𝑡 · 𝑌))
14	11, 13	oveq12d 7378	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)) = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌)))
15	14	eqeq2d 2752	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ 𝑝 = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))))
16	15	rexbidv 3165	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))))
17	16	rabbidv 3400	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))} = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))})
18	17	adantl 483	. . 3 ⊢ (((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))} = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))})
19		sneq 4568	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
20	19	difeq2d 4060	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑃 ∖ {𝑥}) = (𝑃 ∖ {𝑋}))
21	20	adantl 483	. . 3 ⊢ (((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑥 = 𝑋) → (𝑃 ∖ {𝑥}) = (𝑃 ∖ {𝑋}))
22		simpr1 1202	. . 3 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ∈ 𝑃)
23		id 22	. . . . . . . 8 ⊢ (𝑋 ≠ 𝑌 → 𝑋 ≠ 𝑌)
24	23	necomd 2991	. . . . . . 7 ⊢ (𝑋 ≠ 𝑌 → 𝑌 ≠ 𝑋)
25	24	anim2i 624	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ 𝑃 ∧ 𝑌 ≠ 𝑋))
26	25	3adant1 1137	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ 𝑃 ∧ 𝑌 ≠ 𝑋))
27		eldifsn 4722	. . . . 5 ⊢ (𝑌 ∈ (𝑃 ∖ {𝑋}) ↔ (𝑌 ∈ 𝑃 ∧ 𝑌 ≠ 𝑋))
28	26, 27	sylibr 236	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ (𝑃 ∖ {𝑋}))
29	28	adantl 483	. . 3 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → 𝑌 ∈ (𝑃 ∖ {𝑋}))
30	2	ovexi 7394	. . . . 5 ⊢ 𝑃 ∈ V
31	30	rabex 5270	. . . 4 ⊢ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))} ∈ V
32	31	a1i 11	. . 3 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))} ∈ V)
33	9, 18, 21, 22, 29, 32	ovmpodx 7511	. 2 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → (𝑋(𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))})𝑌) = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))})
34	8, 33	eqtrd 2776	1 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∃wrex 3065 {crab 3393 Vcvv 3433 ∖ cdif 3882 {csn 4558 ‘cfv 6489 (class class class)co 7360 ∈ cmpo 7362 ↑_m cmap 8767 Fincfn 8887 ℝcr 11032 1c1 11034 − cmin 11372 +_gcplusg 17215 ·_𝑠 cvsca 17219 ℝ^crrx 25372 Line_Mcline 49232

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-rp 12938 df-fz 13457 df-seq 13959 df-exp 14019 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-0g 17399 df-prds 17405 df-pws 17407 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19094 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-cring 20212 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-subrng 20522 df-subrg 20546 df-drng 20707 df-field 20708 df-sra 21167 df-rgmod 21168 df-cnfld 21352 df-refld 21584 df-dsmm 21711 df-frlm 21726 df-tng 24571 df-tcph 25158 df-rrx 25374 df-line 49234

This theorem is referenced by: rrxlinec 49241

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