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Mirrors > Home > MPE Home > Th. List > pf1f | Structured version Unicode version |
Description: Polynomial functions are functions. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
pf1rcl.q |
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pf1f.b |
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Ref | Expression |
---|---|
pf1f |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2450 |
. 2
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2 | pf1f.b |
. 2
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3 | eqid 2450 |
. 2
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4 | pf1rcl.q |
. . 3
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5 | 4 | pf1rcl 17878 |
. 2
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6 | fvex 5785 |
. . . 4
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7 | 2, 6 | eqeltri 2532 |
. . 3
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8 | 7 | a1i 11 |
. 2
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9 | 2, 4 | pf1subrg 17877 |
. . . 4
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10 | 3 | subrgss 16958 |
. . . 4
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11 | 5, 9, 10 | 3syl 20 |
. . 3
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12 | id 22 |
. . 3
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13 | 11, 12 | sseldd 3441 |
. 2
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14 | 1, 2, 3, 5, 8, 13 | pwselbas 14515 |
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 1709 ax-7 1729 ax-8 1759 ax-9 1761 ax-10 1776 ax-11 1781 ax-12 1793 ax-13 1944 ax-ext 2429 ax-rep 4487 ax-sep 4497 ax-nul 4505 ax-pow 4554 ax-pr 4615 ax-un 6458 ax-inf2 7934 ax-cnex 9425 ax-resscn 9426 ax-1cn 9427 ax-icn 9428 ax-addcl 9429 ax-addrcl 9430 ax-mulcl 9431 ax-mulrcl 9432 ax-mulcom 9433 ax-addass 9434 ax-mulass 9435 ax-distr 9436 ax-i2m1 9437 ax-1ne0 9438 ax-1rid 9439 ax-rnegex 9440 ax-rrecex 9441 ax-cnre 9442 ax-pre-lttri 9443 ax-pre-lttrn 9444 ax-pre-ltadd 9445 ax-pre-mulgt0 9446 |
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 1702 df-eu 2263 df-mo 2264 df-clab 2436 df-cleq 2442 df-clel 2445 df-nfc 2598 df-ne 2643 df-nel 2644 df-ral 2797 df-rex 2798 df-reu 2799 df-rmo 2800 df-rab 2801 df-v 3056 df-sbc 3271 df-csb 3373 df-dif 3415 df-un 3417 df-in 3419 df-ss 3426 df-pss 3428 df-nul 3722 df-if 3876 df-pw 3946 df-sn 3962 df-pr 3964 df-tp 3966 df-op 3968 df-uni 4176 df-int 4213 df-iun 4257 df-iin 4258 df-br 4377 df-opab 4435 df-mpt 4436 df-tr 4470 df-eprel 4716 df-id 4720 df-po 4725 df-so 4726 df-fr 4763 df-se 4764 df-we 4765 df-ord 4806 df-on 4807 df-lim 4808 df-suc 4809 df-xp 4930 df-rel 4931 df-cnv 4932 df-co 4933 df-dm 4934 df-rn 4935 df-res 4936 df-ima 4937 df-iota 5465 df-fun 5504 df-fn 5505 df-f 5506 df-f1 5507 df-fo 5508 df-f1o 5509 df-fv 5510 df-isom 5511 df-riota 6137 df-ov 6179 df-oprab 6180 df-mpt2 6181 df-of 6406 df-ofr 6407 df-om 6563 df-1st 6663 df-2nd 6664 df-supp 6777 df-recs 6918 df-rdg 6952 df-1o 7006 df-2o 7007 df-oadd 7010 df-er 7187 df-map 7302 df-pm 7303 df-ixp 7350 df-en 7397 df-dom 7398 df-sdom 7399 df-fin 7400 df-fsupp 7708 df-sup 7778 df-oi 7811 df-card 8196 df-pnf 9507 df-mnf 9508 df-xr 9509 df-ltxr 9510 df-le 9511 df-sub 9684 df-neg 9685 df-nn 10410 df-2 10467 df-3 10468 df-4 10469 df-5 10470 df-6 10471 df-7 10472 df-8 10473 df-9 10474 df-10 10475 df-n0 10667 df-z 10734 df-dec 10843 df-uz 10949 df-fz 11525 df-fzo 11636 df-seq 11894 df-hash 12191 df-struct 14264 df-ndx 14265 df-slot 14266 df-base 14267 df-sets 14268 df-ress 14269 df-plusg 14339 df-mulr 14340 df-sca 14342 df-vsca 14343 df-ip 14344 df-tset 14345 df-ple 14346 df-ds 14348 df-hom 14350 df-cco 14351 df-0g 14468 df-gsum 14469 df-prds 14474 df-pws 14476 df-mre 14612 df-mrc 14613 df-acs 14615 df-mnd 15503 df-mhm 15552 df-submnd 15553 df-grp 15633 df-minusg 15634 df-sbg 15635 df-mulg 15636 df-subg 15766 df-ghm 15833 df-cntz 15923 df-cmn 16369 df-abl 16370 df-mgp 16683 df-ur 16695 df-srg 16699 df-rng 16739 df-cring 16740 df-rnghom 16898 df-subrg 16955 df-lmod 17042 df-lss 17106 df-lsp 17145 df-assa 17476 df-asp 17477 df-ascl 17478 df-psr 17515 df-mvr 17516 df-mpl 17517 df-opsr 17519 df-evls 17681 df-evl 17682 df-psr1 17729 df-ply1 17731 df-evl1 17846 |
This theorem is referenced by: pf1addcl 17882 pf1mulcl 17883 pf1ind 17884 |
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