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Mirrors > Home > MPE Home > Th. List > abelthlem4 | Structured version Visualization version GIF version |
Description: Lemma for abelth 26190. (Contributed by Mario Carneiro, 31-Mar-2015.) |
Ref | Expression |
---|---|
abelth.1 | β’ (π β π΄:β0βΆβ) |
abelth.2 | β’ (π β seq0( + , π΄) β dom β ) |
abelth.3 | β’ (π β π β β) |
abelth.4 | β’ (π β 0 β€ π) |
abelth.5 | β’ π = {π§ β β β£ (absβ(1 β π§)) β€ (π Β· (1 β (absβπ§)))} |
abelth.6 | β’ πΉ = (π₯ β π β¦ Ξ£π β β0 ((π΄βπ) Β· (π₯βπ))) |
Ref | Expression |
---|---|
abelthlem4 | β’ (π β πΉ:πβΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 12869 | . . 3 β’ β0 = (β€β₯β0) | |
2 | 0zd 12575 | . . 3 β’ ((π β§ π₯ β π) β 0 β β€) | |
3 | fveq2 6891 | . . . . . 6 β’ (π = π β (π΄βπ) = (π΄βπ)) | |
4 | oveq2 7420 | . . . . . 6 β’ (π = π β (π₯βπ) = (π₯βπ)) | |
5 | 3, 4 | oveq12d 7430 | . . . . 5 β’ (π = π β ((π΄βπ) Β· (π₯βπ)) = ((π΄βπ) Β· (π₯βπ))) |
6 | eqid 2731 | . . . . 5 β’ (π β β0 β¦ ((π΄βπ) Β· (π₯βπ))) = (π β β0 β¦ ((π΄βπ) Β· (π₯βπ))) | |
7 | ovex 7445 | . . . . 5 β’ ((π΄βπ) Β· (π₯βπ)) β V | |
8 | 5, 6, 7 | fvmpt 6998 | . . . 4 β’ (π β β0 β ((π β β0 β¦ ((π΄βπ) Β· (π₯βπ)))βπ) = ((π΄βπ) Β· (π₯βπ))) |
9 | 8 | adantl 481 | . . 3 β’ (((π β§ π₯ β π) β§ π β β0) β ((π β β0 β¦ ((π΄βπ) Β· (π₯βπ)))βπ) = ((π΄βπ) Β· (π₯βπ))) |
10 | abelth.1 | . . . . . 6 β’ (π β π΄:β0βΆβ) | |
11 | 10 | adantr 480 | . . . . 5 β’ ((π β§ π₯ β π) β π΄:β0βΆβ) |
12 | 11 | ffvelcdmda 7086 | . . . 4 β’ (((π β§ π₯ β π) β§ π β β0) β (π΄βπ) β β) |
13 | abelth.5 | . . . . . . . 8 β’ π = {π§ β β β£ (absβ(1 β π§)) β€ (π Β· (1 β (absβπ§)))} | |
14 | 13 | ssrab3 4080 | . . . . . . 7 β’ π β β |
15 | 14 | a1i 11 | . . . . . 6 β’ (π β π β β) |
16 | 15 | sselda 3982 | . . . . 5 β’ ((π β§ π₯ β π) β π₯ β β) |
17 | expcl 14050 | . . . . 5 β’ ((π₯ β β β§ π β β0) β (π₯βπ) β β) | |
18 | 16, 17 | sylan 579 | . . . 4 β’ (((π β§ π₯ β π) β§ π β β0) β (π₯βπ) β β) |
19 | 12, 18 | mulcld 11239 | . . 3 β’ (((π β§ π₯ β π) β§ π β β0) β ((π΄βπ) Β· (π₯βπ)) β β) |
20 | abelth.2 | . . . 4 β’ (π β seq0( + , π΄) β dom β ) | |
21 | abelth.3 | . . . 4 β’ (π β π β β) | |
22 | abelth.4 | . . . 4 β’ (π β 0 β€ π) | |
23 | 10, 20, 21, 22, 13 | abelthlem3 26182 | . . 3 β’ ((π β§ π₯ β π) β seq0( + , (π β β0 β¦ ((π΄βπ) Β· (π₯βπ)))) β dom β ) |
24 | 1, 2, 9, 19, 23 | isumcl 15712 | . 2 β’ ((π β§ π₯ β π) β Ξ£π β β0 ((π΄βπ) Β· (π₯βπ)) β β) |
25 | abelth.6 | . 2 β’ πΉ = (π₯ β π β¦ Ξ£π β β0 ((π΄βπ) Β· (π₯βπ))) | |
26 | 24, 25 | fmptd 7115 | 1 β’ (π β πΉ:πβΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 {crab 3431 β wss 3948 class class class wbr 5148 β¦ cmpt 5231 dom cdm 5676 βΆwf 6539 βcfv 6543 (class class class)co 7412 βcc 11112 βcr 11113 0cc0 11114 1c1 11115 + caddc 11117 Β· cmul 11119 β€ cle 11254 β cmin 11449 β0cn0 12477 seqcseq 13971 βcexp 14032 abscabs 15186 β cli 15433 Ξ£csu 15637 |
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-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-inf 9442 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-xadd 13098 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 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-limsup 15420 df-clim 15437 df-rlim 15438 df-sum 15638 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 |
This theorem is referenced by: abelthlem7 26187 abelthlem8 26188 abelthlem9 26189 abelth 26190 |
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