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Mirrors > Home > MPE Home > Th. List > ply1bas | Structured version Visualization version GIF version |
Description: The value of the base set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.) |
Ref | Expression |
---|---|
ply1val.1 | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1val.2 | ⊢ 𝑆 = (PwSer1‘𝑅) |
ply1bas.u | ⊢ 𝑈 = (Base‘𝑃) |
Ref | Expression |
---|---|
ply1bas | ⊢ 𝑈 = (Base‘(1o mPoly 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1bas.u | . 2 ⊢ 𝑈 = (Base‘𝑃) | |
2 | eqid 2823 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
3 | eqid 2823 | . . . 4 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
4 | eqid 2823 | . . . 4 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅)) | |
5 | ply1val.2 | . . . . 5 ⊢ 𝑆 = (PwSer1‘𝑅) | |
6 | eqid 2823 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
7 | 5, 6, 3 | psr1bas2 20360 | . . . 4 ⊢ (Base‘𝑆) = (Base‘(1o mPwSer 𝑅)) |
8 | 2, 3, 4, 7 | mplbasss 20214 | . . 3 ⊢ (Base‘(1o mPoly 𝑅)) ⊆ (Base‘𝑆) |
9 | ply1val.1 | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
10 | 9, 5 | ply1val 20364 | . . . 4 ⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅))) |
11 | 10, 6 | ressbas2 16557 | . . 3 ⊢ ((Base‘(1o mPoly 𝑅)) ⊆ (Base‘𝑆) → (Base‘(1o mPoly 𝑅)) = (Base‘𝑃)) |
12 | 8, 11 | ax-mp 5 | . 2 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘𝑃) |
13 | 1, 12 | eqtr4i 2849 | 1 ⊢ 𝑈 = (Base‘(1o mPoly 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊆ wss 3938 ‘cfv 6357 (class class class)co 7158 1oc1o 8097 Basecbs 16485 mPwSer cmps 20133 mPoly cmpl 20135 PwSer1cps1 20345 Poly1cpl1 20347 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-dec 12102 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-ple 16587 df-psr 20138 df-mpl 20140 df-opsr 20142 df-psr1 20350 df-ply1 20352 |
This theorem is referenced by: ply1lss 20366 ply1subrg 20367 ply1crng 20368 ply1assa 20369 ply1basf 20372 ply1bascl2 20374 vr1cl 20387 ressply1bas2 20398 ressply1add 20400 ressply1mul 20401 ressply1vsca 20402 subrgply1 20403 ply1baspropd 20413 ply1ring 20418 ply1lmod 20422 ply1mpl0 20425 ply1mpl1 20427 subrg1asclcl 20430 subrgvr1cl 20432 coe1add 20434 coe1tm 20443 ply1coe 20466 evls1rhm 20487 evls1sca 20488 evl1rhm 20497 evl1sca 20499 evl1var 20501 evls1var 20503 mpfpf1 20516 pf1mpf 20517 deg1xrf 24677 deg1cl 24679 deg1nn0cl 24684 deg1ldg 24688 deg1leb 24691 deg1val 24692 deg1vscale 24700 deg1vsca 24701 deg1mulle2 24705 deg1le0 24707 fply1 30933 |
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