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Mirrors > Home > MPE Home > Th. List > plyssc | Structured version Visualization version Unicode version |
Description: Every polynomial ring is
contained in the ring of polynomials over
![]() |
Ref | Expression |
---|---|
plyssc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3765 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | sseq1 3455 |
. . 3
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3 | 1, 2 | mpbiri 237 |
. 2
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4 | n0 3743 |
. . 3
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5 | plybss 23160 |
. . . . 5
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6 | ssid 3453 |
. . . . 5
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7 | plyss 23165 |
. . . . 5
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8 | 5, 6, 7 | sylancl 669 |
. . . 4
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9 | 8 | exlimiv 1778 |
. . 3
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10 | 4, 9 | sylbi 199 |
. 2
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11 | 3, 10 | pm2.61ine 2709 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 ax-cnex 9600 ax-resscn 9601 ax-1cn 9602 ax-icn 9603 ax-addcl 9604 ax-addrcl 9605 ax-mulcl 9606 ax-mulrcl 9607 ax-i2m1 9612 ax-1ne0 9613 ax-rrecex 9616 ax-cnre 9617 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-reu 2746 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-uni 4202 df-iun 4283 df-br 4406 df-opab 4465 df-mpt 4466 df-tr 4501 df-eprel 4748 df-id 4752 df-po 4758 df-so 4759 df-fr 4796 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-pred 5383 df-ord 5429 df-on 5430 df-lim 5431 df-suc 5432 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-om 6698 df-1st 6798 df-2nd 6799 df-wrecs 7033 df-recs 7095 df-rdg 7133 df-map 7479 df-nn 10617 df-n0 10877 df-ply 23154 |
This theorem is referenced by: plyaddcl 23186 plymulcl 23187 plysubcl 23188 coeval 23189 coeeu 23191 dgrval 23194 coef3 23198 coeidlem 23203 coemulc 23221 coesub 23223 dgrmulc 23237 dgrsub 23238 dgrcolem1 23239 dgrcolem2 23240 dgrco 23241 coecj 23244 dvply2 23251 dvnply 23253 quotval 23257 quotlem 23265 quotcl2 23267 quotdgr 23268 plyrem 23270 facth 23271 fta1 23273 quotcan 23274 vieta1lem1 23275 vieta1 23277 plyexmo 23278 ftalem7 24017 dgrsub2 36006 |
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