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Mirrors > Home > MPE Home > Th. List > axlowdim1 | Structured version Visualization version GIF version |
Description: The lower dimension axiom for one dimension. In any dimension, there are at least two distinct points. Theorem 3.13 of [Schwabhauser] p. 32, where it is derived from axlowdim2 28486. (Contributed by Scott Fenton, 22-Apr-2013.) |
Ref | Expression |
---|---|
axlowdim1 | β’ (π β β β βπ₯ β (πΌβπ)βπ¦ β (πΌβπ)π₯ β π¦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 11219 | . . . 4 β’ 1 β β | |
2 | 1 | fconst6 6781 | . . 3 β’ ((1...π) Γ {1}):(1...π)βΆβ |
3 | elee 28420 | . . 3 β’ (π β β β (((1...π) Γ {1}) β (πΌβπ) β ((1...π) Γ {1}):(1...π)βΆβ)) | |
4 | 2, 3 | mpbiri 258 | . 2 β’ (π β β β ((1...π) Γ {1}) β (πΌβπ)) |
5 | 0re 11221 | . . . 4 β’ 0 β β | |
6 | 5 | fconst6 6781 | . . 3 β’ ((1...π) Γ {0}):(1...π)βΆβ |
7 | elee 28420 | . . 3 β’ (π β β β (((1...π) Γ {0}) β (πΌβπ) β ((1...π) Γ {0}):(1...π)βΆβ)) | |
8 | 6, 7 | mpbiri 258 | . 2 β’ (π β β β ((1...π) Γ {0}) β (πΌβπ)) |
9 | ax-1ne0 11182 | . . . . . . 7 β’ 1 β 0 | |
10 | 9 | neii 2941 | . . . . . 6 β’ Β¬ 1 = 0 |
11 | 1ex 11215 | . . . . . . 7 β’ 1 β V | |
12 | 11 | sneqr 4841 | . . . . . 6 β’ ({1} = {0} β 1 = 0) |
13 | 10, 12 | mto 196 | . . . . 5 β’ Β¬ {1} = {0} |
14 | elnnuz 12871 | . . . . . . . . 9 β’ (π β β β π β (β€β₯β1)) | |
15 | eluzfz1 13513 | . . . . . . . . 9 β’ (π β (β€β₯β1) β 1 β (1...π)) | |
16 | 14, 15 | sylbi 216 | . . . . . . . 8 β’ (π β β β 1 β (1...π)) |
17 | 16 | ne0d 4335 | . . . . . . 7 β’ (π β β β (1...π) β β ) |
18 | rnxp 6169 | . . . . . . 7 β’ ((1...π) β β β ran ((1...π) Γ {1}) = {1}) | |
19 | 17, 18 | syl 17 | . . . . . 6 β’ (π β β β ran ((1...π) Γ {1}) = {1}) |
20 | rnxp 6169 | . . . . . . 7 β’ ((1...π) β β β ran ((1...π) Γ {0}) = {0}) | |
21 | 17, 20 | syl 17 | . . . . . 6 β’ (π β β β ran ((1...π) Γ {0}) = {0}) |
22 | 19, 21 | eqeq12d 2747 | . . . . 5 β’ (π β β β (ran ((1...π) Γ {1}) = ran ((1...π) Γ {0}) β {1} = {0})) |
23 | 13, 22 | mtbiri 327 | . . . 4 β’ (π β β β Β¬ ran ((1...π) Γ {1}) = ran ((1...π) Γ {0})) |
24 | rneq 5935 | . . . 4 β’ (((1...π) Γ {1}) = ((1...π) Γ {0}) β ran ((1...π) Γ {1}) = ran ((1...π) Γ {0})) | |
25 | 23, 24 | nsyl 140 | . . 3 β’ (π β β β Β¬ ((1...π) Γ {1}) = ((1...π) Γ {0})) |
26 | 25 | neqned 2946 | . 2 β’ (π β β β ((1...π) Γ {1}) β ((1...π) Γ {0})) |
27 | neeq1 3002 | . . 3 β’ (π₯ = ((1...π) Γ {1}) β (π₯ β π¦ β ((1...π) Γ {1}) β π¦)) | |
28 | neeq2 3003 | . . 3 β’ (π¦ = ((1...π) Γ {0}) β (((1...π) Γ {1}) β π¦ β ((1...π) Γ {1}) β ((1...π) Γ {0}))) | |
29 | 27, 28 | rspc2ev 3624 | . 2 β’ ((((1...π) Γ {1}) β (πΌβπ) β§ ((1...π) Γ {0}) β (πΌβπ) β§ ((1...π) Γ {1}) β ((1...π) Γ {0})) β βπ₯ β (πΌβπ)βπ¦ β (πΌβπ)π₯ β π¦) |
30 | 4, 8, 26, 29 | syl3anc 1370 | 1 β’ (π β β β βπ₯ β (πΌβπ)βπ¦ β (πΌβπ)π₯ β π¦) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 β wne 2939 βwrex 3069 β c0 4322 {csn 4628 Γ cxp 5674 ran crn 5677 βΆwf 6539 βcfv 6543 (class class class)co 7412 βcr 11112 0cc0 11113 1c1 11114 βcn 12217 β€β₯cuz 12827 ...cfz 13489 πΌcee 28414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-z 12564 df-uz 12828 df-fz 13490 df-ee 28417 |
This theorem is referenced by: btwndiff 35304 |
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