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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 22141 | . . 3 ⊢ (𝜑 → (+g‘(1o mPwSer 𝑅)) = (+g‘(1o mPwSer 𝑆))) |
5 | eqid 2727 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
6 | eqid 2727 | . . . 4 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
7 | eqid 2727 | . . . 4 ⊢ (+g‘(1o mPoly 𝑅)) = (+g‘(1o mPoly 𝑅)) | |
8 | 5, 6, 7 | mplplusg 21936 | . . 3 ⊢ (+g‘(1o mPoly 𝑅)) = (+g‘(1o mPwSer 𝑅)) |
9 | eqid 2727 | . . . 4 ⊢ (1o mPoly 𝑆) = (1o mPoly 𝑆) | |
10 | eqid 2727 | . . . 4 ⊢ (1o mPwSer 𝑆) = (1o mPwSer 𝑆) | |
11 | eqid 2727 | . . . 4 ⊢ (+g‘(1o mPoly 𝑆)) = (+g‘(1o mPoly 𝑆)) | |
12 | 9, 10, 11 | mplplusg 21936 | . . 3 ⊢ (+g‘(1o mPoly 𝑆)) = (+g‘(1o mPwSer 𝑆)) |
13 | 4, 8, 12 | 3eqtr4g 2792 | . 2 ⊢ (𝜑 → (+g‘(1o mPoly 𝑅)) = (+g‘(1o mPoly 𝑆))) |
14 | eqid 2727 | . . 3 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
15 | eqid 2727 | . . 3 ⊢ (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘𝑅)) | |
16 | 14, 5, 15 | ply1plusg 22129 | . 2 ⊢ (+g‘(Poly1‘𝑅)) = (+g‘(1o mPoly 𝑅)) |
17 | eqid 2727 | . . 3 ⊢ (Poly1‘𝑆) = (Poly1‘𝑆) | |
18 | eqid 2727 | . . 3 ⊢ (+g‘(Poly1‘𝑆)) = (+g‘(Poly1‘𝑆)) | |
19 | 17, 9, 18 | ply1plusg 22129 | . 2 ⊢ (+g‘(Poly1‘𝑆)) = (+g‘(1o mPoly 𝑆)) |
20 | 13, 16, 19 | 3eqtr4g 2792 | 1 ⊢ (𝜑 → (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ‘cfv 6542 (class class class)co 7414 1oc1o 8473 Basecbs 17171 +gcplusg 17224 mPwSer cmps 21824 mPoly cmpl 21826 Poly1cpl1 22083 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-sca 17240 df-vsca 17241 df-tset 17243 df-ple 17244 df-psr 21829 df-mpl 21831 df-opsr 21833 df-psr1 22086 df-ply1 22088 |
This theorem is referenced by: ply1divalg2 26061 |
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