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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 13507 | . . . . . . 7 β’ (π₯ β (1...(πΌ β 1)) β π₯ β β€) | |
2 | 1 | adantl 481 | . . . . . 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 12589 | . . . . . . . 8 β’ (π β π β β€) |
5 | 4 | adantr 480 | . . . . . . 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 12589 | . . . . . . . 8 β’ (π β πΌ β β€) |
8 | 7 | adantr 480 | . . . . . . 7 β’ ((π β§ π₯ β (1...(πΌ β 1))) β πΌ β β€) |
9 | 5, 8 | zsubcld 12675 | . . . . . 6 β’ ((π β§ π₯ β (1...(πΌ β 1))) β (π β πΌ) β β€) |
10 | 2, 9 | zaddcld 12674 | . . . . 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 7109 | . . . 4 β’ (π β πΆ:(1...(πΌ β 1))βΆβ€) |
13 | 12 | ffnd 6712 | . . 3 β’ (π β πΆ Fn (1...(πΌ β 1))) |
14 | elfzelz 13507 | . . . . . . 7 β’ (π₯ β (πΌ...(π β 1)) β π₯ β β€) | |
15 | 14 | adantl 481 | . . . . . 6 β’ ((π β§ π₯ β (πΌ...(π β 1))) β π₯ β β€) |
16 | 1zzd 12597 | . . . . . . 7 β’ ((π β§ π₯ β (πΌ...(π β 1))) β 1 β β€) | |
17 | 7 | adantr 480 | . . . . . . 7 β’ ((π β§ π₯ β (πΌ...(π β 1))) β πΌ β β€) |
18 | 16, 17 | zsubcld 12675 | . . . . . 6 β’ ((π β§ π₯ β (πΌ...(π β 1))) β (1 β πΌ) β β€) |
19 | 15, 18 | zaddcld 12674 | . . . . 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 7109 | . . . 4 β’ (π β π·:(πΌ...(π β 1))βΆβ€) |
22 | 21 | ffnd 6712 | . . 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 41568 | . . . . . 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 494 | . . . . 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 6658 | . . 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 6594 | . . 3 β’ ((π β β β§ π β β) β {β¨π, πβ©} Fn {π}) | |
30 | 3, 3, 29 | syl2anc 583 | . 2 β’ (π β {β¨π, πβ©} Fn {π}) |
31 | 28, 30 | jca 511 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βͺ cun 3941 β© cin 3942 β c0 4317 ifcif 4523 {csn 4623 β¨cop 4629 class class class wbr 5141 β¦ cmpt 5224 Fn wfn 6532 (class class class)co 7405 1c1 11113 + caddc 11115 < clt 11252 β€ cle 11253 β cmin 11448 βcn 12216 β€cz 12562 ...cfz 13490 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 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 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 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-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-fz 13491 |
This theorem is referenced by: metakunt20 41570 metakunt21 41571 metakunt22 41572 metakunt25 41575 |
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