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| Mirrors > Home > MPE Home > Th. List > evl1addd | Structured version Visualization version GIF version | ||
| Description: Polynomial evaluation builder for addition of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.) |
| Ref | Expression |
|---|---|
| evl1addd.q | ⊢ 𝑂 = (eval1‘𝑅) |
| evl1addd.p | ⊢ 𝑃 = (Poly1‘𝑅) |
| evl1addd.b | ⊢ 𝐵 = (Base‘𝑅) |
| evl1addd.u | ⊢ 𝑈 = (Base‘𝑃) |
| evl1addd.1 | ⊢ (𝜑 → 𝑅 ∈ CRing) |
| evl1addd.2 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| evl1addd.3 | ⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉)) |
| evl1addd.4 | ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊)) |
| evl1addd.g | ⊢ ✚ = (+g‘𝑃) |
| evl1addd.a | ⊢ + = (+g‘𝑅) |
| Ref | Expression |
|---|---|
| evl1addd | ⊢ (𝜑 → ((𝑀 ✚ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | evl1addd.1 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
| 2 | evl1addd.q | . . . . . . 7 ⊢ 𝑂 = (eval1‘𝑅) | |
| 3 | evl1addd.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 4 | eqid 2737 | . . . . . . 7 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
| 5 | evl1addd.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
| 6 | 2, 3, 4, 5 | evl1rhm 22336 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
| 7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
| 8 | rhmghm 20484 | . . . . 5 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵)) → 𝑂 ∈ (𝑃 GrpHom (𝑅 ↑s 𝐵))) | |
| 9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑂 ∈ (𝑃 GrpHom (𝑅 ↑s 𝐵))) |
| 10 | ghmgrp1 19236 | . . . 4 ⊢ (𝑂 ∈ (𝑃 GrpHom (𝑅 ↑s 𝐵)) → 𝑃 ∈ Grp) | |
| 11 | 9, 10 | syl 17 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Grp) |
| 12 | evl1addd.3 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉)) | |
| 13 | 12 | simpld 494 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑈) |
| 14 | evl1addd.4 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊)) | |
| 15 | 14 | simpld 494 | . . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑈) |
| 16 | evl1addd.u | . . . 4 ⊢ 𝑈 = (Base‘𝑃) | |
| 17 | evl1addd.g | . . . 4 ⊢ ✚ = (+g‘𝑃) | |
| 18 | 16, 17 | grpcl 18959 | . . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑀 ✚ 𝑁) ∈ 𝑈) |
| 19 | 11, 13, 15, 18 | syl3anc 1373 | . 2 ⊢ (𝜑 → (𝑀 ✚ 𝑁) ∈ 𝑈) |
| 20 | eqid 2737 | . . . . . . 7 ⊢ (+g‘(𝑅 ↑s 𝐵)) = (+g‘(𝑅 ↑s 𝐵)) | |
| 21 | 16, 17, 20 | ghmlin 19239 | . . . . . 6 ⊢ ((𝑂 ∈ (𝑃 GrpHom (𝑅 ↑s 𝐵)) ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑂‘(𝑀 ✚ 𝑁)) = ((𝑂‘𝑀)(+g‘(𝑅 ↑s 𝐵))(𝑂‘𝑁))) |
| 22 | 9, 13, 15, 21 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 ✚ 𝑁)) = ((𝑂‘𝑀)(+g‘(𝑅 ↑s 𝐵))(𝑂‘𝑁))) |
| 23 | eqid 2737 | . . . . . 6 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
| 24 | 5 | fvexi 6920 | . . . . . . 7 ⊢ 𝐵 ∈ V |
| 25 | 24 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
| 26 | 16, 23 | rhmf 20485 | . . . . . . . 8 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵)) → 𝑂:𝑈⟶(Base‘(𝑅 ↑s 𝐵))) |
| 27 | 7, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑂:𝑈⟶(Base‘(𝑅 ↑s 𝐵))) |
| 28 | 27, 13 | ffvelcdmd 7105 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑s 𝐵))) |
| 29 | 27, 15 | ffvelcdmd 7105 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑁) ∈ (Base‘(𝑅 ↑s 𝐵))) |
| 30 | evl1addd.a | . . . . . 6 ⊢ + = (+g‘𝑅) | |
| 31 | 4, 23, 1, 25, 28, 29, 30, 20 | pwsplusgval 17535 | . . . . 5 ⊢ (𝜑 → ((𝑂‘𝑀)(+g‘(𝑅 ↑s 𝐵))(𝑂‘𝑁)) = ((𝑂‘𝑀) ∘f + (𝑂‘𝑁))) |
| 32 | 22, 31 | eqtrd 2777 | . . . 4 ⊢ (𝜑 → (𝑂‘(𝑀 ✚ 𝑁)) = ((𝑂‘𝑀) ∘f + (𝑂‘𝑁))) |
| 33 | 32 | fveq1d 6908 | . . 3 ⊢ (𝜑 → ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (((𝑂‘𝑀) ∘f + (𝑂‘𝑁))‘𝑌)) |
| 34 | 4, 5, 23, 1, 25, 28 | pwselbas 17534 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵) |
| 35 | 34 | ffnd 6737 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑀) Fn 𝐵) |
| 36 | 4, 5, 23, 1, 25, 29 | pwselbas 17534 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑁):𝐵⟶𝐵) |
| 37 | 36 | ffnd 6737 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑁) Fn 𝐵) |
| 38 | evl1addd.2 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 39 | fnfvof 7714 | . . . 4 ⊢ ((((𝑂‘𝑀) Fn 𝐵 ∧ (𝑂‘𝑁) Fn 𝐵) ∧ (𝐵 ∈ V ∧ 𝑌 ∈ 𝐵)) → (((𝑂‘𝑀) ∘f + (𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌) + ((𝑂‘𝑁)‘𝑌))) | |
| 40 | 35, 37, 25, 38, 39 | syl22anc 839 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀) ∘f + (𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌) + ((𝑂‘𝑁)‘𝑌))) |
| 41 | 12 | simprd 495 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) = 𝑉) |
| 42 | 14 | simprd 495 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑁)‘𝑌) = 𝑊) |
| 43 | 41, 42 | oveq12d 7449 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀)‘𝑌) + ((𝑂‘𝑁)‘𝑌)) = (𝑉 + 𝑊)) |
| 44 | 33, 40, 43 | 3eqtrd 2781 | . 2 ⊢ (𝜑 → ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊)) |
| 45 | 19, 44 | jca 511 | 1 ⊢ (𝜑 → ((𝑀 ✚ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 Basecbs 17247 +gcplusg 17297 ↑s cpws 17491 Grpcgrp 18951 GrpHom cghm 19230 CRingccrg 20231 RingHom crh 20469 Poly1cpl1 22178 eval1ce1 22318 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-srg 20184 df-ring 20232 df-cring 20233 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-lsp 20970 df-assa 21873 df-asp 21874 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-evls 22098 df-evl 22099 df-psr1 22181 df-ply1 22183 df-evl1 22320 |
| This theorem is referenced by: evl1gsumdlem 22360 evls1addd 22375 aks6d1c1p2 42110 aks6d1c1p3 42111 aks6d1c5lem1 42137 aks6d1c5lem2 42139 aks5lem3a 42190 |
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