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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 2819 | . . . . . . 7 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
5 | evl1addd.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 2, 3, 4, 5 | evl1rhm 20487 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
8 | rhmghm 19469 | . . . . 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 497 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑈) |
14 | evl1addd.4 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊)) | |
15 | 14 | simpld 497 | . . 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 1366 | . 2 ⊢ (𝜑 → (𝑀 ✚ 𝑁) ∈ 𝑈) |
20 | eqid 2819 | . . . . . . 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 1366 | . . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 ✚ 𝑁)) = ((𝑂‘𝑀)(+g‘(𝑅 ↑s 𝐵))(𝑂‘𝑁))) |
23 | eqid 2819 | . . . . . 6 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
24 | 5 | fvexi 6677 | . . . . . . 7 ⊢ 𝐵 ∈ V |
25 | 24 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
26 | 16, 23 | rhmf 19470 | . . . . . . . 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 6845 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑s 𝐵))) |
29 | 27, 15 | ffvelrnd 6845 | . . . . . 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 2854 | . . . 4 ⊢ (𝜑 → (𝑂‘(𝑀 ✚ 𝑁)) = ((𝑂‘𝑀) ∘f + (𝑂‘𝑁))) |
33 | 32 | fveq1d 6665 | . . 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 6508 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑀) Fn 𝐵) |
36 | 4, 5, 23, 1, 25, 29 | pwselbas 16754 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑁):𝐵⟶𝐵) |
37 | 36 | ffnd 6508 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑁) Fn 𝐵) |
38 | evl1addd.2 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
39 | fnfvof 7415 | . . . 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 836 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀) ∘f + (𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌) + ((𝑂‘𝑁)‘𝑌))) |
41 | 12 | simprd 498 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) = 𝑉) |
42 | 14 | simprd 498 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑁)‘𝑌) = 𝑊) |
43 | 41, 42 | oveq12d 7166 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀)‘𝑌) + ((𝑂‘𝑁)‘𝑌)) = (𝑉 + 𝑊)) |
44 | 33, 40, 43 | 3eqtrd 2858 | . 2 ⊢ (𝜑 → ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊)) |
45 | 19, 44 | jca 514 | 1 ⊢ (𝜑 → ((𝑀 ✚ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 Vcvv 3493 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 ∘f cof 7399 Basecbs 16475 +gcplusg 16557 ↑s cpws 16712 Grpcgrp 18095 GrpHom cghm 18347 CRingccrg 19290 RingHom crh 19456 Poly1cpl1 20337 eval1ce1 20469 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-ofr 7402 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-sup 8898 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-fz 12885 df-fzo 13026 df-seq 13362 df-hash 13683 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 19232 df-ur 19244 df-srg 19248 df-ring 19291 df-cring 19292 df-rnghom 19459 df-subrg 19525 df-lmod 19628 df-lss 19696 df-lsp 19736 df-assa 20077 df-asp 20078 df-ascl 20079 df-psr 20128 df-mvr 20129 df-mpl 20130 df-opsr 20132 df-evls 20278 df-evl 20279 df-psr1 20340 df-ply1 20342 df-evl1 20471 |
This theorem is referenced by: evl1gsumdlem 20511 |
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