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Mirrors > Home > MPE Home > Th. List > tglowdim1i | Structured version Visualization version GIF version |
Description: Lower dimension axiom for one dimension. (Contributed by Thierry Arnoux, 28-May-2019.) |
Ref | Expression |
---|---|
tglowdim1.p | β’ π = (BaseβπΊ) |
tglowdim1.d | β’ β = (distβπΊ) |
tglowdim1.i | β’ πΌ = (ItvβπΊ) |
tglowdim1.g | β’ (π β πΊ β TarskiG) |
tglowdim1.1 | β’ (π β 2 β€ (β―βπ)) |
tglowdim1i.1 | β’ (π β π β π) |
Ref | Expression |
---|---|
tglowdim1i | β’ (π β βπ¦ β π π β π¦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglowdim1.p | . . . . 5 β’ π = (BaseβπΊ) | |
2 | tglowdim1.d | . . . . 5 β’ β = (distβπΊ) | |
3 | tglowdim1.i | . . . . 5 β’ πΌ = (ItvβπΊ) | |
4 | tglowdim1.g | . . . . . 6 β’ (π β πΊ β TarskiG) | |
5 | 4 | adantr 481 | . . . . 5 β’ ((π β§ βπ¦ β π π = π¦) β πΊ β TarskiG) |
6 | tglowdim1.1 | . . . . . 6 β’ (π β 2 β€ (β―βπ)) | |
7 | 6 | adantr 481 | . . . . 5 β’ ((π β§ βπ¦ β π π = π¦) β 2 β€ (β―βπ)) |
8 | 1, 2, 3, 5, 7 | tglowdim1 27442 | . . . 4 β’ ((π β§ βπ¦ β π π = π¦) β βπ β π βπ β π π β π) |
9 | eqeq2 2748 | . . . . . . . . 9 β’ (π¦ = π β (π = π¦ β π = π)) | |
10 | simpllr 774 | . . . . . . . . 9 β’ ((((π β§ βπ¦ β π π = π¦) β§ π β π) β§ π β π) β βπ¦ β π π = π¦) | |
11 | simplr 767 | . . . . . . . . 9 β’ ((((π β§ βπ¦ β π π = π¦) β§ π β π) β§ π β π) β π β π) | |
12 | 9, 10, 11 | rspcdva 3582 | . . . . . . . 8 β’ ((((π β§ βπ¦ β π π = π¦) β§ π β π) β§ π β π) β π = π) |
13 | eqeq2 2748 | . . . . . . . . . 10 β’ (π¦ = π β (π = π¦ β π = π)) | |
14 | 13 | rspccva 3580 | . . . . . . . . 9 β’ ((βπ¦ β π π = π¦ β§ π β π) β π = π) |
15 | 14 | ad4ant24 752 | . . . . . . . 8 β’ ((((π β§ βπ¦ β π π = π¦) β§ π β π) β§ π β π) β π = π) |
16 | 12, 15 | eqtr3d 2778 | . . . . . . 7 β’ ((((π β§ βπ¦ β π π = π¦) β§ π β π) β§ π β π) β π = π) |
17 | nne 2947 | . . . . . . 7 β’ (Β¬ π β π β π = π) | |
18 | 16, 17 | sylibr 233 | . . . . . 6 β’ ((((π β§ βπ¦ β π π = π¦) β§ π β π) β§ π β π) β Β¬ π β π) |
19 | 18 | nrexdv 3146 | . . . . 5 β’ (((π β§ βπ¦ β π π = π¦) β§ π β π) β Β¬ βπ β π π β π) |
20 | 19 | nrexdv 3146 | . . . 4 β’ ((π β§ βπ¦ β π π = π¦) β Β¬ βπ β π βπ β π π β π) |
21 | 8, 20 | pm2.65da 815 | . . 3 β’ (π β Β¬ βπ¦ β π π = π¦) |
22 | rexnal 3103 | . . 3 β’ (βπ¦ β π Β¬ π = π¦ β Β¬ βπ¦ β π π = π¦) | |
23 | 21, 22 | sylibr 233 | . 2 β’ (π β βπ¦ β π Β¬ π = π¦) |
24 | df-ne 2944 | . . 3 β’ (π β π¦ β Β¬ π = π¦) | |
25 | 24 | rexbii 3097 | . 2 β’ (βπ¦ β π π β π¦ β βπ¦ β π Β¬ π = π¦) |
26 | 23, 25 | sylibr 233 | 1 β’ (π β βπ¦ β π π β π¦) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β wne 2943 βwral 3064 βwrex 3073 class class class wbr 5105 βcfv 6496 β€ cle 11190 2c2 12208 β―chash 14230 Basecbs 17083 distcds 17142 TarskiGcstrkg 27369 Itvcitv 27375 |
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 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-fz 13425 df-hash 14231 |
This theorem is referenced by: colline 27591 |
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