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Mirrors > Home > MPE Home > Th. List > opptgdim2 | Structured version Visualization version GIF version |
Description: If two points opposite to a line exist, dimension must be 2 or more. (Contributed by Thierry Arnoux, 3-Mar-2020.) |
Ref | Expression |
---|---|
hpg.p | β’ π = (BaseβπΊ) |
hpg.d | β’ β = (distβπΊ) |
hpg.i | β’ πΌ = (ItvβπΊ) |
hpg.o | β’ π = {β¨π, πβ© β£ ((π β (π β π·) β§ π β (π β π·)) β§ βπ‘ β π· π‘ β (ππΌπ))} |
opphl.l | β’ πΏ = (LineGβπΊ) |
opphl.d | β’ (π β π· β ran πΏ) |
opphl.g | β’ (π β πΊ β TarskiG) |
oppcom.a | β’ (π β π΄ β π) |
oppcom.b | β’ (π β π΅ β π) |
oppcom.o | β’ (π β π΄ππ΅) |
Ref | Expression |
---|---|
opptgdim2 | β’ (π β πΊDimTarskiGβ₯2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hpg.p | . . 3 β’ π = (BaseβπΊ) | |
2 | opphl.l | . . 3 β’ πΏ = (LineGβπΊ) | |
3 | hpg.i | . . 3 β’ πΌ = (ItvβπΊ) | |
4 | opphl.g | . . . 4 β’ (π β πΊ β TarskiG) | |
5 | 4 | ad3antrrr 728 | . . 3 β’ ((((π β§ π₯ β π) β§ π¦ β π) β§ (π· = (π₯πΏπ¦) β§ π₯ β π¦)) β πΊ β TarskiG) |
6 | simpllr 774 | . . 3 β’ ((((π β§ π₯ β π) β§ π¦ β π) β§ (π· = (π₯πΏπ¦) β§ π₯ β π¦)) β π₯ β π) | |
7 | simplr 767 | . . 3 β’ ((((π β§ π₯ β π) β§ π¦ β π) β§ (π· = (π₯πΏπ¦) β§ π₯ β π¦)) β π¦ β π) | |
8 | oppcom.a | . . . 4 β’ (π β π΄ β π) | |
9 | 8 | ad3antrrr 728 | . . 3 β’ ((((π β§ π₯ β π) β§ π¦ β π) β§ (π· = (π₯πΏπ¦) β§ π₯ β π¦)) β π΄ β π) |
10 | hpg.d | . . . . . . 7 β’ β = (distβπΊ) | |
11 | hpg.o | . . . . . . 7 β’ π = {β¨π, πβ© β£ ((π β (π β π·) β§ π β (π β π·)) β§ βπ‘ β π· π‘ β (ππΌπ))} | |
12 | opphl.d | . . . . . . 7 β’ (π β π· β ran πΏ) | |
13 | oppcom.b | . . . . . . 7 β’ (π β π΅ β π) | |
14 | oppcom.o | . . . . . . 7 β’ (π β π΄ππ΅) | |
15 | 1, 10, 3, 11, 2, 12, 4, 8, 13, 14 | oppne1 28247 | . . . . . 6 β’ (π β Β¬ π΄ β π·) |
16 | 15 | ad3antrrr 728 | . . . . 5 β’ ((((π β§ π₯ β π) β§ π¦ β π) β§ (π· = (π₯πΏπ¦) β§ π₯ β π¦)) β Β¬ π΄ β π·) |
17 | simprl 769 | . . . . 5 β’ ((((π β§ π₯ β π) β§ π¦ β π) β§ (π· = (π₯πΏπ¦) β§ π₯ β π¦)) β π· = (π₯πΏπ¦)) | |
18 | 16, 17 | neleqtrd 2855 | . . . 4 β’ ((((π β§ π₯ β π) β§ π¦ β π) β§ (π· = (π₯πΏπ¦) β§ π₯ β π¦)) β Β¬ π΄ β (π₯πΏπ¦)) |
19 | simprr 771 | . . . . 5 β’ ((((π β§ π₯ β π) β§ π¦ β π) β§ (π· = (π₯πΏπ¦) β§ π₯ β π¦)) β π₯ β π¦) | |
20 | 19 | neneqd 2945 | . . . 4 β’ ((((π β§ π₯ β π) β§ π¦ β π) β§ (π· = (π₯πΏπ¦) β§ π₯ β π¦)) β Β¬ π₯ = π¦) |
21 | ioran 982 | . . . 4 β’ (Β¬ (π΄ β (π₯πΏπ¦) β¨ π₯ = π¦) β (Β¬ π΄ β (π₯πΏπ¦) β§ Β¬ π₯ = π¦)) | |
22 | 18, 20, 21 | sylanbrc 583 | . . 3 β’ ((((π β§ π₯ β π) β§ π¦ β π) β§ (π· = (π₯πΏπ¦) β§ π₯ β π¦)) β Β¬ (π΄ β (π₯πΏπ¦) β¨ π₯ = π¦)) |
23 | 1, 2, 3, 5, 6, 7, 9, 22 | ncoltgdim2 28071 | . 2 β’ ((((π β§ π₯ β π) β§ π¦ β π) β§ (π· = (π₯πΏπ¦) β§ π₯ β π¦)) β πΊDimTarskiGβ₯2) |
24 | 1, 3, 2, 4, 12 | tgisline 28133 | . 2 β’ (π β βπ₯ β π βπ¦ β π (π· = (π₯πΏπ¦) β§ π₯ β π¦)) |
25 | 23, 24 | r19.29vva 3213 | 1 β’ (π β πΊDimTarskiGβ₯2) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 β¨ wo 845 = wceq 1541 β wcel 2106 β wne 2940 βwrex 3070 β cdif 3945 class class class wbr 5148 {copab 5210 ran crn 5677 βcfv 6543 (class class class)co 7411 2c2 12271 Basecbs 17148 distcds 17210 TarskiGcstrkg 27933 DimTarskiGβ₯cstrkgld 27937 Itvcitv 27939 LineGclng 27940 |
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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-trkgc 27954 df-trkgcb 27956 df-trkgld 27958 df-trkg 27959 |
This theorem is referenced by: opphllem5 28257 opphl 28260 |
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