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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 2820 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
3 | eqid 2820 | . . . 4 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
4 | eqid 2820 | . . . 4 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅)) | |
5 | ply1val.2 | . . . . 5 ⊢ 𝑆 = (PwSer1‘𝑅) | |
6 | eqid 2820 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
7 | 5, 6, 3 | psr1bas2 20353 | . . . 4 ⊢ (Base‘𝑆) = (Base‘(1o mPwSer 𝑅)) |
8 | 2, 3, 4, 7 | mplbasss 20207 | . . 3 ⊢ (Base‘(1o mPoly 𝑅)) ⊆ (Base‘𝑆) |
9 | ply1val.1 | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
10 | 9, 5 | ply1val 20357 | . . . 4 ⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅))) |
11 | 10, 6 | ressbas2 16550 | . . 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 2846 | 1 ⊢ 𝑈 = (Base‘(1o mPoly 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ⊆ wss 3929 ‘cfv 6348 (class class class)co 7149 1oc1o 8088 Basecbs 16478 mPwSer cmps 20126 mPoly cmpl 20128 PwSer1cps1 20338 Poly1cpl1 20340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-dec 12093 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-ple 16580 df-psr 20131 df-mpl 20133 df-opsr 20135 df-psr1 20343 df-ply1 20345 |
This theorem is referenced by: ply1lss 20359 ply1subrg 20360 ply1crng 20361 ply1assa 20362 ply1basf 20365 ply1bascl2 20367 vr1cl 20380 ressply1bas2 20391 ressply1add 20393 ressply1mul 20394 ressply1vsca 20395 subrgply1 20396 ply1baspropd 20406 ply1ring 20411 ply1lmod 20415 ply1mpl0 20418 ply1mpl1 20420 subrg1asclcl 20423 subrgvr1cl 20425 coe1add 20427 coe1tm 20436 ply1coe 20459 evls1rhm 20480 evls1sca 20481 evl1rhm 20490 evl1sca 20492 evl1var 20494 evls1var 20496 mpfpf1 20509 pf1mpf 20510 deg1xrf 24673 deg1cl 24675 deg1nn0cl 24680 deg1ldg 24684 deg1leb 24687 deg1val 24688 deg1vscale 24696 deg1vsca 24697 deg1mulle2 24701 deg1le0 24703 fply1 30952 |
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