Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ply1plusg | Structured version Visualization version GIF version |
Description: Value of addition in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.) |
Ref | Expression |
---|---|
ply1plusg.y | ⊢ 𝑌 = (Poly1‘𝑅) |
ply1plusg.s | ⊢ 𝑆 = (1o mPoly 𝑅) |
ply1plusg.p | ⊢ + = (+g‘𝑌) |
Ref | Expression |
---|---|
ply1plusg | ⊢ + = (+g‘𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1plusg.p | . 2 ⊢ + = (+g‘𝑌) | |
2 | ply1plusg.s | . . . 4 ⊢ 𝑆 = (1o mPoly 𝑅) | |
3 | eqid 2736 | . . . 4 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
4 | eqid 2736 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
5 | 2, 3, 4 | mplplusg 21497 | . . 3 ⊢ (+g‘𝑆) = (+g‘(1o mPwSer 𝑅)) |
6 | eqid 2736 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
7 | eqid 2736 | . . . 4 ⊢ (+g‘(PwSer1‘𝑅)) = (+g‘(PwSer1‘𝑅)) | |
8 | 6, 3, 7 | psr1plusg 21499 | . . 3 ⊢ (+g‘(PwSer1‘𝑅)) = (+g‘(1o mPwSer 𝑅)) |
9 | fvex 6838 | . . . 4 ⊢ (Base‘(1o mPoly 𝑅)) ∈ V | |
10 | ply1plusg.y | . . . . . 6 ⊢ 𝑌 = (Poly1‘𝑅) | |
11 | 10, 6 | ply1val 21471 | . . . . 5 ⊢ 𝑌 = ((PwSer1‘𝑅) ↾s (Base‘(1o mPoly 𝑅))) |
12 | 11, 7 | ressplusg 17097 | . . . 4 ⊢ ((Base‘(1o mPoly 𝑅)) ∈ V → (+g‘(PwSer1‘𝑅)) = (+g‘𝑌)) |
13 | 9, 12 | ax-mp 5 | . . 3 ⊢ (+g‘(PwSer1‘𝑅)) = (+g‘𝑌) |
14 | 5, 8, 13 | 3eqtr2i 2770 | . 2 ⊢ (+g‘𝑆) = (+g‘𝑌) |
15 | 1, 14 | eqtr4i 2767 | 1 ⊢ + = (+g‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 Vcvv 3441 ‘cfv 6479 (class class class)co 7337 1oc1o 8360 Basecbs 17009 +gcplusg 17059 mPwSer cmps 21213 mPoly cmpl 21215 PwSer1cps1 21452 Poly1cpl1 21454 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-3 12138 df-4 12139 df-5 12140 df-6 12141 df-7 12142 df-8 12143 df-9 12144 df-dec 12539 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-ple 17079 df-psr 21218 df-mpl 21220 df-opsr 21222 df-psr1 21457 df-ply1 21459 |
This theorem is referenced by: ressply1add 21507 subrgply1 21510 ply1plusgfvi 21519 ply1plusgpropd 21521 ply1mpl0 21532 coe1add 21541 ply1coe 21573 evls1rhm 21594 evl1rhm 21604 deg1addle 25372 |
Copyright terms: Public domain | W3C validator |