rrxline - Mathbox for Alexander van der Vekens

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Theorem rrxline 46138

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 46137	. . . 4 ⊢ (𝐼 ∈ Fin → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
7	6	oveqd 7324	. . 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 2737	. . 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 7323	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((1 − 𝑡) · 𝑥) = ((1 − 𝑡) · 𝑋))
12		simpr 486	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
13	12	oveq2d 7323	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑡 · 𝑦) = (𝑡 · 𝑌))
14	11, 13	oveq12d 7325	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)) = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌)))
15	14	eqeq2d 2747	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ 𝑝 = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))))
16	15	rexbidv 3172	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))))
17	16	rabbidv 3421	. . . 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 4575	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
20	19	difeq2d 4063	. . . 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 1194	. . 3 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ∈ 𝑃)
23		id 22	. . . . . . . 8 ⊢ (𝑋 ≠ 𝑌 → 𝑋 ≠ 𝑌)
24	23	necomd 2997	. . . . . . 7 ⊢ (𝑋 ≠ 𝑌 → 𝑌 ≠ 𝑋)
25	24	anim2i 618	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ 𝑃 ∧ 𝑌 ≠ 𝑋))
26	25	3adant1 1130	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ 𝑃 ∧ 𝑌 ≠ 𝑋))
27		eldifsn 4726	. . . . 5 ⊢ (𝑌 ∈ (𝑃 ∖ {𝑋}) ↔ (𝑌 ∈ 𝑃 ∧ 𝑌 ≠ 𝑋))
28	26, 27	sylibr 233	. . . 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 7341	. . . . 5 ⊢ 𝑃 ∈ V
31	30	rabex 5265	. . . 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 7456	. 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 1087 = wceq 1539 ∈ wcel 2104 ≠ wne 2941 ∃wrex 3071 {crab 3284 Vcvv 3437 ∖ cdif 3889 {csn 4565 ‘cfv 6458 (class class class)co 7307 ∈ cmpo 7309 ↑_m cmap 8646 Fincfn 8764 ℝcr 10916 1c1 10918 − cmin 11251 +_gcplusg 17007 ·_𝑠 cvsca 17011 ℝ^crrx 24592 Line_Mcline 46131

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 ax-pre-sup 10995 ax-addf 10996 ax-mulf 10997

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-supp 8009 df-tpos 8073 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-map 8648 df-ixp 8717 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-fsupp 9173 df-sup 9245 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-div 11679 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-7 12087 df-8 12088 df-9 12089 df-n0 12280 df-z 12366 df-dec 12484 df-uz 12629 df-rp 12777 df-fz 13286 df-seq 13768 df-exp 13829 df-cj 14855 df-re 14856 df-im 14857 df-sqrt 14991 df-abs 14992 df-struct 16893 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-plusg 17020 df-mulr 17021 df-starv 17022 df-sca 17023 df-vsca 17024 df-ip 17025 df-tset 17026 df-ple 17027 df-ds 17029 df-unif 17030 df-hom 17031 df-cco 17032 df-0g 17197 df-prds 17203 df-pws 17205 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-grp 18625 df-minusg 18626 df-sbg 18627 df-subg 18797 df-cmn 19433 df-mgp 19766 df-ur 19783 df-ring 19830 df-cring 19831 df-oppr 19907 df-dvdsr 19928 df-unit 19929 df-invr 19959 df-dvr 19970 df-drng 20038 df-field 20039 df-subrg 20067 df-sra 20479 df-rgmod 20480 df-cnfld 20643 df-refld 20855 df-dsmm 20984 df-frlm 20999 df-tng 23785 df-tcph 24378 df-rrx 24594 df-line 46133

This theorem is referenced by: rrxlinec 46140

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