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Mirrors > Home > MPE Home > Th. List > Mathboxes > sticksstones14 | Structured version Visualization version GIF version |
Description: Sticks and stones with definitions as hypotheses. (Contributed by metakunt, 7-Oct-2024.) |
Ref | Expression |
---|---|
sticksstones14.1 | β’ (π β π β β0) |
sticksstones14.2 | β’ (π β πΎ β β0) |
sticksstones14.3 | β’ πΉ = (π β π΄ β¦ (π β (1...πΎ) β¦ (π + Ξ£π β (1...π)(πβπ)))) |
sticksstones14.4 | β’ πΊ = (π β π΅ β¦ if(πΎ = 0, {β¨1, πβ©}, (π β (1...(πΎ + 1)) β¦ if(π = (πΎ + 1), ((π + πΎ) β (πβπΎ)), if(π = 1, ((πβ1) β 1), (((πβπ) β (πβ(π β 1))) β 1)))))) |
sticksstones14.5 | β’ π΄ = {π β£ (π:(1...(πΎ + 1))βΆβ0 β§ Ξ£π β (1...(πΎ + 1))(πβπ) = π)} |
sticksstones14.6 | β’ π΅ = {π β£ (π:(1...πΎ)βΆ(1...(π + πΎ)) β§ βπ₯ β (1...πΎ)βπ¦ β (1...πΎ)(π₯ < π¦ β (πβπ₯) < (πβπ¦)))} |
Ref | Expression |
---|---|
sticksstones14 | β’ (π β (β―βπ΄) = ((π + πΎ)CπΎ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sticksstones14.5 | . . . . 5 β’ π΄ = {π β£ (π:(1...(πΎ + 1))βΆβ0 β§ Ξ£π β (1...(πΎ + 1))(πβπ) = π)} | |
2 | 1 | a1i 11 | . . . 4 β’ (π β π΄ = {π β£ (π:(1...(πΎ + 1))βΆβ0 β§ Ξ£π β (1...(πΎ + 1))(πβπ) = π)}) |
3 | simpl 482 | . . . . . . 7 β’ ((π:(1...(πΎ + 1))βΆβ0 β§ Ξ£π β (1...(πΎ + 1))(πβπ) = π) β π:(1...(πΎ + 1))βΆβ0) | |
4 | 3 | a1i 11 | . . . . . 6 β’ (π β ((π:(1...(πΎ + 1))βΆβ0 β§ Ξ£π β (1...(πΎ + 1))(πβπ) = π) β π:(1...(πΎ + 1))βΆβ0)) |
5 | 4 | ss2abdv 4056 | . . . . 5 β’ (π β {π β£ (π:(1...(πΎ + 1))βΆβ0 β§ Ξ£π β (1...(πΎ + 1))(πβπ) = π)} β {π β£ π:(1...(πΎ + 1))βΆβ0}) |
6 | fzfid 13962 | . . . . . 6 β’ (π β (1...(πΎ + 1)) β Fin) | |
7 | nn0ex 12500 | . . . . . . 7 β’ β0 β V | |
8 | 7 | a1i 11 | . . . . . 6 β’ (π β β0 β V) |
9 | mapex 8842 | . . . . . 6 β’ (((1...(πΎ + 1)) β Fin β§ β0 β V) β {π β£ π:(1...(πΎ + 1))βΆβ0} β V) | |
10 | 6, 8, 9 | syl2anc 583 | . . . . 5 β’ (π β {π β£ π:(1...(πΎ + 1))βΆβ0} β V) |
11 | ssexg 5317 | . . . . 5 β’ (({π β£ (π:(1...(πΎ + 1))βΆβ0 β§ Ξ£π β (1...(πΎ + 1))(πβπ) = π)} β {π β£ π:(1...(πΎ + 1))βΆβ0} β§ {π β£ π:(1...(πΎ + 1))βΆβ0} β V) β {π β£ (π:(1...(πΎ + 1))βΆβ0 β§ Ξ£π β (1...(πΎ + 1))(πβπ) = π)} β V) | |
12 | 5, 10, 11 | syl2anc 583 | . . . 4 β’ (π β {π β£ (π:(1...(πΎ + 1))βΆβ0 β§ Ξ£π β (1...(πΎ + 1))(πβπ) = π)} β V) |
13 | 2, 12 | eqeltrd 2828 | . . 3 β’ (π β π΄ β V) |
14 | sticksstones14.1 | . . . 4 β’ (π β π β β0) | |
15 | sticksstones14.2 | . . . 4 β’ (π β πΎ β β0) | |
16 | sticksstones14.3 | . . . 4 β’ πΉ = (π β π΄ β¦ (π β (1...πΎ) β¦ (π + Ξ£π β (1...π)(πβπ)))) | |
17 | sticksstones14.4 | . . . 4 β’ πΊ = (π β π΅ β¦ if(πΎ = 0, {β¨1, πβ©}, (π β (1...(πΎ + 1)) β¦ if(π = (πΎ + 1), ((π + πΎ) β (πβπΎ)), if(π = 1, ((πβ1) β 1), (((πβπ) β (πβ(π β 1))) β 1)))))) | |
18 | sticksstones14.6 | . . . 4 β’ π΅ = {π β£ (π:(1...πΎ)βΆ(1...(π + πΎ)) β§ βπ₯ β (1...πΎ)βπ¦ β (1...πΎ)(π₯ < π¦ β (πβπ₯) < (πβπ¦)))} | |
19 | 14, 15, 16, 17, 1, 18 | sticksstones13 41563 | . . 3 β’ (π β πΉ:π΄β1-1-ontoβπ΅) |
20 | 13, 19 | hasheqf1od 14336 | . 2 β’ (π β (β―βπ΄) = (β―βπ΅)) |
21 | 14, 15 | nn0addcld 12558 | . . 3 β’ (π β (π + πΎ) β β0) |
22 | 21, 15, 18 | sticksstones5 41554 | . 2 β’ (π β (β―βπ΅) = ((π + πΎ)CπΎ)) |
23 | 20, 22 | eqtrd 2767 | 1 β’ (π β (β―βπ΄) = ((π + πΎ)CπΎ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 {cab 2704 βwral 3056 Vcvv 3469 β wss 3944 ifcif 4524 {csn 4624 β¨cop 4630 class class class wbr 5142 β¦ cmpt 5225 βΆwf 6538 βcfv 6542 (class class class)co 7414 Fincfn 8955 0cc0 11130 1c1 11131 + caddc 11133 < clt 11270 β cmin 11466 β0cn0 12494 ...cfz 13508 Ccbc 14285 β―chash 14313 Ξ£csu 15656 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-inf 9458 df-oi 9525 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-ico 13354 df-fz 13509 df-fzo 13652 df-seq 13991 df-exp 14051 df-fac 14257 df-bc 14286 df-hash 14314 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-clim 15456 df-sum 15657 |
This theorem is referenced by: sticksstones15 41565 |
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