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Mirrors > Home > MPE Home > Th. List > psrelbas | Structured version Unicode version |
Description: An element of the set of power series is a function on the coefficients. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Ref | Expression |
---|---|
psrbas.s |
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psrbas.k |
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psrbas.d |
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psrbas.b |
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psrelbas.x |
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Ref | Expression |
---|---|
psrelbas |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrelbas.x |
. . 3
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2 | psrbas.s |
. . . 4
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3 | psrbas.k |
. . . 4
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4 | psrbas.d |
. . . 4
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5 | psrbas.b |
. . . 4
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6 | reldmpsr 17550 |
. . . . . . 7
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7 | 6, 2, 5 | elbasov 14339 |
. . . . . 6
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8 | 1, 7 | syl 16 |
. . . . 5
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9 | 8 | simpld 459 |
. . . 4
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10 | 2, 3, 4, 5, 9 | psrbas 17570 |
. . 3
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11 | 1, 10 | eleqtrd 2544 |
. 2
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12 | fvex 5808 |
. . . 4
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13 | 3, 12 | eqeltri 2538 |
. . 3
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14 | ovex 6224 |
. . . . 5
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15 | 14 | rabex 4550 |
. . . 4
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16 | 4, 15 | eqeltri 2538 |
. . 3
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17 | 13, 16 | elmap 7350 |
. 2
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18 | 11, 17 | sylib 196 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4510 ax-sep 4520 ax-nul 4528 ax-pow 4577 ax-pr 4638 ax-un 6481 ax-cnex 9448 ax-resscn 9449 ax-1cn 9450 ax-icn 9451 ax-addcl 9452 ax-addrcl 9453 ax-mulcl 9454 ax-mulrcl 9455 ax-mulcom 9456 ax-addass 9457 ax-mulass 9458 ax-distr 9459 ax-i2m1 9460 ax-1ne0 9461 ax-1rid 9462 ax-rnegex 9463 ax-rrecex 9464 ax-cnre 9465 ax-pre-lttri 9466 ax-pre-lttrn 9467 ax-pre-ltadd 9468 ax-pre-mulgt0 9469 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2649 df-nel 2650 df-ral 2803 df-rex 2804 df-reu 2805 df-rab 2807 df-v 3078 df-sbc 3293 df-csb 3395 df-dif 3438 df-un 3440 df-in 3442 df-ss 3449 df-pss 3451 df-nul 3745 df-if 3899 df-pw 3969 df-sn 3985 df-pr 3987 df-tp 3989 df-op 3991 df-uni 4199 df-int 4236 df-iun 4280 df-br 4400 df-opab 4458 df-mpt 4459 df-tr 4493 df-eprel 4739 df-id 4743 df-po 4748 df-so 4749 df-fr 4786 df-we 4788 df-ord 4829 df-on 4830 df-lim 4831 df-suc 4832 df-xp 4953 df-rel 4954 df-cnv 4955 df-co 4956 df-dm 4957 df-rn 4958 df-res 4959 df-ima 4960 df-iota 5488 df-fun 5527 df-fn 5528 df-f 5529 df-f1 5530 df-fo 5531 df-f1o 5532 df-fv 5533 df-riota 6160 df-ov 6202 df-oprab 6203 df-mpt2 6204 df-of 6429 df-om 6586 df-1st 6686 df-2nd 6687 df-supp 6800 df-recs 6941 df-rdg 6975 df-1o 7029 df-oadd 7033 df-er 7210 df-map 7325 df-en 7420 df-dom 7421 df-sdom 7422 df-fin 7423 df-fsupp 7731 df-pnf 9530 df-mnf 9531 df-xr 9532 df-ltxr 9533 df-le 9534 df-sub 9707 df-neg 9708 df-nn 10433 df-2 10490 df-3 10491 df-4 10492 df-5 10493 df-6 10494 df-7 10495 df-8 10496 df-9 10497 df-n0 10690 df-z 10757 df-uz 10972 df-fz 11554 df-struct 14293 df-ndx 14294 df-slot 14295 df-base 14296 df-plusg 14369 df-mulr 14370 df-sca 14372 df-vsca 14373 df-tset 14375 df-psr 17545 |
This theorem is referenced by: psrelbasfun 17573 psraddcl 17576 psrmulcllem 17580 psrvscaval 17585 psrvscacl 17586 psr0lid 17588 psrnegcl 17589 psrlinv 17590 psrgrp 17591 psrlmod 17594 psrlidm 17596 psrlidmOLD 17597 psrridm 17598 psrridmOLD 17599 psrass1 17600 psrdi 17601 psrdir 17602 psrass23l 17603 psrcom 17604 psrass23 17605 resspsrmul 17612 mplvalOLD 17625 mplelf 17632 mplsubglem 17633 mpllsslem 17634 mplsubglemOLD 17635 mpllsslemOLD 17636 mplsubrglem 17640 mplsubrglemOLD 17641 mvrcl 17651 subrgasclcl 17704 psrplusgpropd 17813 psropprmul 17815 |
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