![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ply1plusgpropd | Structured version Visualization version GIF version |
Description: Property deduction for univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
Ref | Expression |
---|---|
ply1baspropd.b1 | ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) |
ply1baspropd.b2 | ⊢ (𝜑 → 𝐵 = (Base‘𝑆)) |
ply1baspropd.p | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦)) |
Ref | Expression |
---|---|
ply1plusgpropd | ⊢ (𝜑 → (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1baspropd.b1 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) | |
2 | ply1baspropd.b2 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝑆)) | |
3 | ply1baspropd.p | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦)) | |
4 | 1, 2, 3 | psrplusgpropd 22161 | . . 3 ⊢ (𝜑 → (+g‘(1o mPwSer 𝑅)) = (+g‘(1o mPwSer 𝑆))) |
5 | eqid 2728 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
6 | eqid 2728 | . . . 4 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
7 | eqid 2728 | . . . 4 ⊢ (+g‘(1o mPoly 𝑅)) = (+g‘(1o mPoly 𝑅)) | |
8 | 5, 6, 7 | mplplusg 21956 | . . 3 ⊢ (+g‘(1o mPoly 𝑅)) = (+g‘(1o mPwSer 𝑅)) |
9 | eqid 2728 | . . . 4 ⊢ (1o mPoly 𝑆) = (1o mPoly 𝑆) | |
10 | eqid 2728 | . . . 4 ⊢ (1o mPwSer 𝑆) = (1o mPwSer 𝑆) | |
11 | eqid 2728 | . . . 4 ⊢ (+g‘(1o mPoly 𝑆)) = (+g‘(1o mPoly 𝑆)) | |
12 | 9, 10, 11 | mplplusg 21956 | . . 3 ⊢ (+g‘(1o mPoly 𝑆)) = (+g‘(1o mPwSer 𝑆)) |
13 | 4, 8, 12 | 3eqtr4g 2793 | . 2 ⊢ (𝜑 → (+g‘(1o mPoly 𝑅)) = (+g‘(1o mPoly 𝑆))) |
14 | eqid 2728 | . . 3 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
15 | eqid 2728 | . . 3 ⊢ (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘𝑅)) | |
16 | 14, 5, 15 | ply1plusg 22149 | . 2 ⊢ (+g‘(Poly1‘𝑅)) = (+g‘(1o mPoly 𝑅)) |
17 | eqid 2728 | . . 3 ⊢ (Poly1‘𝑆) = (Poly1‘𝑆) | |
18 | eqid 2728 | . . 3 ⊢ (+g‘(Poly1‘𝑆)) = (+g‘(Poly1‘𝑆)) | |
19 | 17, 9, 18 | ply1plusg 22149 | . 2 ⊢ (+g‘(Poly1‘𝑆)) = (+g‘(1o mPoly 𝑆)) |
20 | 13, 16, 19 | 3eqtr4g 2793 | 1 ⊢ (𝜑 → (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 1oc1o 8486 Basecbs 17187 +gcplusg 17240 mPwSer cmps 21844 mPoly cmpl 21846 Poly1cpl1 22103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-tset 17259 df-ple 17260 df-psr 21849 df-mpl 21851 df-opsr 21853 df-psr1 22106 df-ply1 22108 |
This theorem is referenced by: ply1divalg2 26094 |
Copyright terms: Public domain | W3C validator |