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Mirrors > Home > MPE Home > Th. List > dvfsumrlimf | Structured version Visualization version GIF version |
Description: Lemma for dvfsumrlim 25784. (Contributed by Mario Carneiro, 18-May-2016.) |
Ref | Expression |
---|---|
dvfsum.s | β’ π = (π(,)+β) |
dvfsum.z | β’ π = (β€β₯βπ) |
dvfsum.m | β’ (π β π β β€) |
dvfsum.d | β’ (π β π· β β) |
dvfsum.md | β’ (π β π β€ (π· + 1)) |
dvfsum.t | β’ (π β π β β) |
dvfsum.a | β’ ((π β§ π₯ β π) β π΄ β β) |
dvfsum.b1 | β’ ((π β§ π₯ β π) β π΅ β π) |
dvfsum.b2 | β’ ((π β§ π₯ β π) β π΅ β β) |
dvfsum.b3 | β’ (π β (β D (π₯ β π β¦ π΄)) = (π₯ β π β¦ π΅)) |
dvfsum.c | β’ (π₯ = π β π΅ = πΆ) |
dvfsumrlimf.g | β’ πΊ = (π₯ β π β¦ (Ξ£π β (π...(ββπ₯))πΆ β π΄)) |
Ref | Expression |
---|---|
dvfsumrlimf | β’ (π β πΊ:πβΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzfid 13943 | . . . 4 β’ ((π β§ π₯ β π) β (π...(ββπ₯)) β Fin) | |
2 | dvfsum.b2 | . . . . . . 7 β’ ((π β§ π₯ β π) β π΅ β β) | |
3 | 2 | ralrimiva 3145 | . . . . . 6 β’ (π β βπ₯ β π π΅ β β) |
4 | 3 | adantr 480 | . . . . 5 β’ ((π β§ π₯ β π) β βπ₯ β π π΅ β β) |
5 | elfzuz 13502 | . . . . . 6 β’ (π β (π...(ββπ₯)) β π β (β€β₯βπ)) | |
6 | dvfsum.z | . . . . . 6 β’ π = (β€β₯βπ) | |
7 | 5, 6 | eleqtrrdi 2843 | . . . . 5 β’ (π β (π...(ββπ₯)) β π β π) |
8 | dvfsum.c | . . . . . . 7 β’ (π₯ = π β π΅ = πΆ) | |
9 | 8 | eleq1d 2817 | . . . . . 6 β’ (π₯ = π β (π΅ β β β πΆ β β)) |
10 | 9 | rspccva 3611 | . . . . 5 β’ ((βπ₯ β π π΅ β β β§ π β π) β πΆ β β) |
11 | 4, 7, 10 | syl2an 595 | . . . 4 β’ (((π β§ π₯ β π) β§ π β (π...(ββπ₯))) β πΆ β β) |
12 | 1, 11 | fsumrecl 15685 | . . 3 β’ ((π β§ π₯ β π) β Ξ£π β (π...(ββπ₯))πΆ β β) |
13 | dvfsum.a | . . 3 β’ ((π β§ π₯ β π) β π΄ β β) | |
14 | 12, 13 | resubcld 11647 | . 2 β’ ((π β§ π₯ β π) β (Ξ£π β (π...(ββπ₯))πΆ β π΄) β β) |
15 | dvfsumrlimf.g | . 2 β’ πΊ = (π₯ β π β¦ (Ξ£π β (π...(ββπ₯))πΆ β π΄)) | |
16 | 14, 15 | fmptd 7115 | 1 β’ (π β πΊ:πβΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 βwral 3060 class class class wbr 5148 β¦ cmpt 5231 βΆwf 6539 βcfv 6543 (class class class)co 7412 βcr 11113 1c1 11115 + caddc 11117 +βcpnf 11250 β€ cle 11254 β cmin 11449 β€cz 12563 β€β₯cuz 12827 (,)cioo 13329 ...cfz 13489 βcfl 13760 Ξ£csu 15637 D cdv 25613 |
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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 |
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-rmo 3375 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-int 4951 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-se 5632 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-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-oi 9509 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-sum 15638 |
This theorem is referenced by: dvfsumrlim 25784 dvfsumrlim2 25785 dvfsumrlim3 25786 |
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