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Mirrors > Home > MPE Home > Th. List > mpladd | Structured version Visualization version GIF version |
Description: The addition operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
mpladd.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mpladd.b | ⊢ 𝐵 = (Base‘𝑃) |
mpladd.a | ⊢ + = (+g‘𝑅) |
mpladd.g | ⊢ ✚ = (+g‘𝑃) |
mpladd.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
mpladd.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
mpladd | ⊢ (𝜑 → (𝑋 ✚ 𝑌) = (𝑋 ∘𝑓 + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2772 | . 2 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
2 | eqid 2772 | . 2 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
3 | mpladd.a | . 2 ⊢ + = (+g‘𝑅) | |
4 | mpladd.g | . . 3 ⊢ ✚ = (+g‘𝑃) | |
5 | mpladd.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑃) | |
6 | 5 | fvexi 6507 | . . . 4 ⊢ 𝐵 ∈ V |
7 | mpladd.p | . . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
8 | 7, 1, 5 | mplval2 19915 | . . . . 5 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾s 𝐵) |
9 | eqid 2772 | . . . . 5 ⊢ (+g‘(𝐼 mPwSer 𝑅)) = (+g‘(𝐼 mPwSer 𝑅)) | |
10 | 8, 9 | ressplusg 16458 | . . . 4 ⊢ (𝐵 ∈ V → (+g‘(𝐼 mPwSer 𝑅)) = (+g‘𝑃)) |
11 | 6, 10 | ax-mp 5 | . . 3 ⊢ (+g‘(𝐼 mPwSer 𝑅)) = (+g‘𝑃) |
12 | 4, 11 | eqtr4i 2799 | . 2 ⊢ ✚ = (+g‘(𝐼 mPwSer 𝑅)) |
13 | 7, 1, 5, 2 | mplbasss 19916 | . . 3 ⊢ 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅)) |
14 | mpladd.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
15 | 13, 14 | sseldi 3852 | . 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘(𝐼 mPwSer 𝑅))) |
16 | mpladd.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
17 | 13, 16 | sseldi 3852 | . 2 ⊢ (𝜑 → 𝑌 ∈ (Base‘(𝐼 mPwSer 𝑅))) |
18 | 1, 2, 3, 12, 15, 17 | psradd 19866 | 1 ⊢ (𝜑 → (𝑋 ✚ 𝑌) = (𝑋 ∘𝑓 + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2048 Vcvv 3409 ‘cfv 6182 (class class class)co 6970 ∘𝑓 cof 7219 Basecbs 16329 +gcplusg 16411 mPwSer cmps 19835 mPoly cmpl 19837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7494 df-2nd 7495 df-supp 7627 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-fsupp 8621 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-uz 12052 df-fz 12702 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-sca 16427 df-vsca 16428 df-tset 16430 df-psr 19840 df-mpl 19842 |
This theorem is referenced by: mplcoe1 19949 evlslem1 19998 mhpaddcl 20034 coe1add 20125 mdegaddle 24361 |
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