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Mirrors > Home > MPE Home > Th. List > Mathboxes > metakunt19 | Structured version Visualization version GIF version |
Description: Domains on restrictions of functions. (Contributed by metakunt, 28-May-2024.) |
Ref | Expression |
---|---|
metakunt19.1 | β’ (π β π β β) |
metakunt19.2 | β’ (π β πΌ β β) |
metakunt19.3 | β’ (π β πΌ β€ π) |
metakunt19.4 | β’ π΅ = (π₯ β (1...π) β¦ if(π₯ = π, π, if(π₯ < πΌ, (π₯ + (π β πΌ)), (π₯ + (1 β πΌ))))) |
metakunt19.5 | β’ πΆ = (π₯ β (1...(πΌ β 1)) β¦ (π₯ + (π β πΌ))) |
metakunt19.6 | β’ π· = (π₯ β (πΌ...(π β 1)) β¦ (π₯ + (1 β πΌ))) |
Ref | Expression |
---|---|
metakunt19 | β’ (π β ((πΆ Fn (1...(πΌ β 1)) β§ π· Fn (πΌ...(π β 1)) β§ (πΆ βͺ π·) Fn ((1...(πΌ β 1)) βͺ (πΌ...(π β 1)))) β§ {β¨π, πβ©} Fn {π})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzelz 13533 | . . . . . . 7 β’ (π₯ β (1...(πΌ β 1)) β π₯ β β€) | |
2 | 1 | adantl 480 | . . . . . 6 β’ ((π β§ π₯ β (1...(πΌ β 1))) β π₯ β β€) |
3 | metakunt19.1 | . . . . . . . . 9 β’ (π β π β β) | |
4 | 3 | nnzd 12615 | . . . . . . . 8 β’ (π β π β β€) |
5 | 4 | adantr 479 | . . . . . . 7 β’ ((π β§ π₯ β (1...(πΌ β 1))) β π β β€) |
6 | metakunt19.2 | . . . . . . . . 9 β’ (π β πΌ β β) | |
7 | 6 | nnzd 12615 | . . . . . . . 8 β’ (π β πΌ β β€) |
8 | 7 | adantr 479 | . . . . . . 7 β’ ((π β§ π₯ β (1...(πΌ β 1))) β πΌ β β€) |
9 | 5, 8 | zsubcld 12701 | . . . . . 6 β’ ((π β§ π₯ β (1...(πΌ β 1))) β (π β πΌ) β β€) |
10 | 2, 9 | zaddcld 12700 | . . . . 5 β’ ((π β§ π₯ β (1...(πΌ β 1))) β (π₯ + (π β πΌ)) β β€) |
11 | metakunt19.5 | . . . . 5 β’ πΆ = (π₯ β (1...(πΌ β 1)) β¦ (π₯ + (π β πΌ))) | |
12 | 10, 11 | fmptd 7119 | . . . 4 β’ (π β πΆ:(1...(πΌ β 1))βΆβ€) |
13 | 12 | ffnd 6718 | . . 3 β’ (π β πΆ Fn (1...(πΌ β 1))) |
14 | elfzelz 13533 | . . . . . . 7 β’ (π₯ β (πΌ...(π β 1)) β π₯ β β€) | |
15 | 14 | adantl 480 | . . . . . 6 β’ ((π β§ π₯ β (πΌ...(π β 1))) β π₯ β β€) |
16 | 1zzd 12623 | . . . . . . 7 β’ ((π β§ π₯ β (πΌ...(π β 1))) β 1 β β€) | |
17 | 7 | adantr 479 | . . . . . . 7 β’ ((π β§ π₯ β (πΌ...(π β 1))) β πΌ β β€) |
18 | 16, 17 | zsubcld 12701 | . . . . . 6 β’ ((π β§ π₯ β (πΌ...(π β 1))) β (1 β πΌ) β β€) |
19 | 15, 18 | zaddcld 12700 | . . . . 5 β’ ((π β§ π₯ β (πΌ...(π β 1))) β (π₯ + (1 β πΌ)) β β€) |
20 | metakunt19.6 | . . . . 5 β’ π· = (π₯ β (πΌ...(π β 1)) β¦ (π₯ + (1 β πΌ))) | |
21 | 19, 20 | fmptd 7119 | . . . 4 β’ (π β π·:(πΌ...(π β 1))βΆβ€) |
22 | 21 | ffnd 6718 | . . 3 β’ (π β π· Fn (πΌ...(π β 1))) |
23 | metakunt19.3 | . . . . . . 7 β’ (π β πΌ β€ π) | |
24 | 3, 6, 23 | metakunt18 41730 | . . . . . 6 β’ (π β ((((1...(πΌ β 1)) β© (πΌ...(π β 1))) = β β§ ((1...(πΌ β 1)) β© {π}) = β β§ ((πΌ...(π β 1)) β© {π}) = β ) β§ (((((π β πΌ) + 1)...(π β 1)) β© (1...(π β πΌ))) = β β§ ((((π β πΌ) + 1)...(π β 1)) β© {π}) = β β§ ((1...(π β πΌ)) β© {π}) = β ))) |
25 | 24 | simpld 493 | . . . . 5 β’ (π β (((1...(πΌ β 1)) β© (πΌ...(π β 1))) = β β§ ((1...(πΌ β 1)) β© {π}) = β β§ ((πΌ...(π β 1)) β© {π}) = β )) |
26 | 25 | simp1d 1139 | . . . 4 β’ (π β ((1...(πΌ β 1)) β© (πΌ...(π β 1))) = β ) |
27 | 13, 22, 26 | fnund 6664 | . . 3 β’ (π β (πΆ βͺ π·) Fn ((1...(πΌ β 1)) βͺ (πΌ...(π β 1)))) |
28 | 13, 22, 27 | 3jca 1125 | . 2 β’ (π β (πΆ Fn (1...(πΌ β 1)) β§ π· Fn (πΌ...(π β 1)) β§ (πΆ βͺ π·) Fn ((1...(πΌ β 1)) βͺ (πΌ...(π β 1))))) |
29 | fnsng 6600 | . . 3 β’ ((π β β β§ π β β) β {β¨π, πβ©} Fn {π}) | |
30 | 3, 3, 29 | syl2anc 582 | . 2 β’ (π β {β¨π, πβ©} Fn {π}) |
31 | 28, 30 | jca 510 | 1 β’ (π β ((πΆ Fn (1...(πΌ β 1)) β§ π· Fn (πΌ...(π β 1)) β§ (πΆ βͺ π·) Fn ((1...(πΌ β 1)) βͺ (πΌ...(π β 1)))) β§ {β¨π, πβ©} Fn {π})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βͺ cun 3937 β© cin 3938 β c0 4318 ifcif 4524 {csn 4624 β¨cop 4630 class class class wbr 5143 β¦ cmpt 5226 Fn wfn 6538 (class class class)co 7416 1c1 11139 + caddc 11141 < clt 11278 β€ cle 11279 β cmin 11474 βcn 12242 β€cz 12588 ...cfz 13516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-fz 13517 |
This theorem is referenced by: metakunt20 41732 metakunt21 41733 metakunt22 41734 metakunt25 41737 |
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