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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 2798 | . . . . . . 7 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
5 | evl1addd.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 2, 3, 4, 5 | evl1rhm 20956 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
8 | rhmghm 19473 | . . . . 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 18352 | . . . 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 498 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑈) |
14 | evl1addd.4 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊)) | |
15 | 14 | simpld 498 | . . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑈) |
16 | evl1addd.u | . . . 4 ⊢ 𝑈 = (Base‘𝑃) | |
17 | evl1addd.g | . . . 4 ⊢ ✚ = (+g‘𝑃) | |
18 | 16, 17 | grpcl 18103 | . . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑀 ✚ 𝑁) ∈ 𝑈) |
19 | 11, 13, 15, 18 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝑀 ✚ 𝑁) ∈ 𝑈) |
20 | eqid 2798 | . . . . . . 7 ⊢ (+g‘(𝑅 ↑s 𝐵)) = (+g‘(𝑅 ↑s 𝐵)) | |
21 | 16, 17, 20 | ghmlin 18355 | . . . . . 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 1368 | . . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 ✚ 𝑁)) = ((𝑂‘𝑀)(+g‘(𝑅 ↑s 𝐵))(𝑂‘𝑁))) |
23 | eqid 2798 | . . . . . 6 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
24 | 5 | fvexi 6659 | . . . . . . 7 ⊢ 𝐵 ∈ V |
25 | 24 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
26 | 16, 23 | rhmf 19474 | . . . . . . . 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 | ffvelrnd 6829 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑s 𝐵))) |
29 | 27, 15 | ffvelrnd 6829 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑁) ∈ (Base‘(𝑅 ↑s 𝐵))) |
30 | evl1addd.a | . . . . . 6 ⊢ + = (+g‘𝑅) | |
31 | 4, 23, 1, 25, 28, 29, 30, 20 | pwsplusgval 16755 | . . . . 5 ⊢ (𝜑 → ((𝑂‘𝑀)(+g‘(𝑅 ↑s 𝐵))(𝑂‘𝑁)) = ((𝑂‘𝑀) ∘f + (𝑂‘𝑁))) |
32 | 22, 31 | eqtrd 2833 | . . . 4 ⊢ (𝜑 → (𝑂‘(𝑀 ✚ 𝑁)) = ((𝑂‘𝑀) ∘f + (𝑂‘𝑁))) |
33 | 32 | fveq1d 6647 | . . 3 ⊢ (𝜑 → ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (((𝑂‘𝑀) ∘f + (𝑂‘𝑁))‘𝑌)) |
34 | 4, 5, 23, 1, 25, 28 | pwselbas 16754 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵) |
35 | 34 | ffnd 6488 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑀) Fn 𝐵) |
36 | 4, 5, 23, 1, 25, 29 | pwselbas 16754 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑁):𝐵⟶𝐵) |
37 | 36 | ffnd 6488 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑁) Fn 𝐵) |
38 | evl1addd.2 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
39 | fnfvof 7403 | . . . 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 837 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀) ∘f + (𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌) + ((𝑂‘𝑁)‘𝑌))) |
41 | 12 | simprd 499 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) = 𝑉) |
42 | 14 | simprd 499 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑁)‘𝑌) = 𝑊) |
43 | 41, 42 | oveq12d 7153 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀)‘𝑌) + ((𝑂‘𝑁)‘𝑌)) = (𝑉 + 𝑊)) |
44 | 33, 40, 43 | 3eqtrd 2837 | . 2 ⊢ (𝜑 → ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊)) |
45 | 19, 44 | jca 515 | 1 ⊢ (𝜑 → ((𝑀 ✚ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 Basecbs 16475 +gcplusg 16557 ↑s cpws 16712 Grpcgrp 18095 GrpHom cghm 18347 CRingccrg 19291 RingHom crh 19460 Poly1cpl1 20806 eval1ce1 20938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-srg 19249 df-ring 19292 df-cring 19293 df-rnghom 19463 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-assa 20542 df-asp 20543 df-ascl 20544 df-psr 20594 df-mvr 20595 df-mpl 20596 df-opsr 20598 df-evls 20745 df-evl 20746 df-psr1 20809 df-ply1 20811 df-evl1 20940 |
This theorem is referenced by: evl1gsumdlem 20980 |
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