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Mirrors > Home > MPE Home > Th. List > Mathboxes > evladdval | Structured version Visualization version GIF version |
Description: Polynomial evaluation builder for addition. (Contributed by SN, 9-Feb-2025.) |
Ref | Expression |
---|---|
evladdval.q | ⊢ 𝑄 = (𝐼 eval 𝑆) |
evladdval.p | ⊢ 𝑃 = (𝐼 mPoly 𝑆) |
evladdval.k | ⊢ 𝐾 = (Base‘𝑆) |
evladdval.b | ⊢ 𝐵 = (Base‘𝑃) |
evladdval.g | ⊢ ✚ = (+g‘𝑃) |
evladdval.f | ⊢ + = (+g‘𝑆) |
evladdval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑍) |
evladdval.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evladdval.a | ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) |
evladdval.m | ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉)) |
evladdval.n | ⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ ((𝑄‘𝑁)‘𝐴) = 𝑊)) |
Ref | Expression |
---|---|
evladdval | ⊢ (𝜑 → ((𝑀 ✚ 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 ✚ 𝑁))‘𝐴) = (𝑉 + 𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evladdval.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
2 | evladdval.g | . . 3 ⊢ ✚ = (+g‘𝑃) | |
3 | evladdval.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑍) | |
4 | evladdval.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
5 | evladdval.q | . . . . . . 7 ⊢ 𝑄 = (𝐼 eval 𝑆) | |
6 | evladdval.k | . . . . . . 7 ⊢ 𝐾 = (Base‘𝑆) | |
7 | evladdval.p | . . . . . . 7 ⊢ 𝑃 = (𝐼 mPoly 𝑆) | |
8 | eqid 2732 | . . . . . . 7 ⊢ (𝑆 ↑s (𝐾 ↑m 𝐼)) = (𝑆 ↑s (𝐾 ↑m 𝐼)) | |
9 | 5, 6, 7, 8 | evlrhm 21658 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing) → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼)))) |
10 | 3, 4, 9 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼)))) |
11 | rhmghm 20261 | . . . . 5 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼))) → 𝑄 ∈ (𝑃 GrpHom (𝑆 ↑s (𝐾 ↑m 𝐼)))) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑃 GrpHom (𝑆 ↑s (𝐾 ↑m 𝐼)))) |
13 | ghmgrp1 19093 | . . . 4 ⊢ (𝑄 ∈ (𝑃 GrpHom (𝑆 ↑s (𝐾 ↑m 𝐼))) → 𝑃 ∈ Grp) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Grp) |
15 | evladdval.m | . . . 4 ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉)) | |
16 | 15 | simpld 495 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝐵) |
17 | evladdval.n | . . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ ((𝑄‘𝑁)‘𝐴) = 𝑊)) | |
18 | 17 | simpld 495 | . . 3 ⊢ (𝜑 → 𝑁 ∈ 𝐵) |
19 | 1, 2, 14, 16, 18 | grpcld 18832 | . 2 ⊢ (𝜑 → (𝑀 ✚ 𝑁) ∈ 𝐵) |
20 | eqid 2732 | . . . . . . 7 ⊢ (+g‘(𝑆 ↑s (𝐾 ↑m 𝐼))) = (+g‘(𝑆 ↑s (𝐾 ↑m 𝐼))) | |
21 | 1, 2, 20 | ghmlin 19096 | . . . . . 6 ⊢ ((𝑄 ∈ (𝑃 GrpHom (𝑆 ↑s (𝐾 ↑m 𝐼))) ∧ 𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵) → (𝑄‘(𝑀 ✚ 𝑁)) = ((𝑄‘𝑀)(+g‘(𝑆 ↑s (𝐾 ↑m 𝐼)))(𝑄‘𝑁))) |
22 | 12, 16, 18, 21 | syl3anc 1371 | . . . . 5 ⊢ (𝜑 → (𝑄‘(𝑀 ✚ 𝑁)) = ((𝑄‘𝑀)(+g‘(𝑆 ↑s (𝐾 ↑m 𝐼)))(𝑄‘𝑁))) |
23 | eqid 2732 | . . . . . 6 ⊢ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) = (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) | |
24 | ovexd 7443 | . . . . . 6 ⊢ (𝜑 → (𝐾 ↑m 𝐼) ∈ V) | |
25 | 1, 23 | rhmf 20262 | . . . . . . . 8 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼))) → 𝑄:𝐵⟶(Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
26 | 10, 25 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑄:𝐵⟶(Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
27 | 26, 16 | ffvelcdmd 7087 | . . . . . 6 ⊢ (𝜑 → (𝑄‘𝑀) ∈ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
28 | 26, 18 | ffvelcdmd 7087 | . . . . . 6 ⊢ (𝜑 → (𝑄‘𝑁) ∈ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
29 | evladdval.f | . . . . . 6 ⊢ + = (+g‘𝑆) | |
30 | 8, 23, 4, 24, 27, 28, 29, 20 | pwsplusgval 17435 | . . . . 5 ⊢ (𝜑 → ((𝑄‘𝑀)(+g‘(𝑆 ↑s (𝐾 ↑m 𝐼)))(𝑄‘𝑁)) = ((𝑄‘𝑀) ∘f + (𝑄‘𝑁))) |
31 | 22, 30 | eqtrd 2772 | . . . 4 ⊢ (𝜑 → (𝑄‘(𝑀 ✚ 𝑁)) = ((𝑄‘𝑀) ∘f + (𝑄‘𝑁))) |
32 | 31 | fveq1d 6893 | . . 3 ⊢ (𝜑 → ((𝑄‘(𝑀 ✚ 𝑁))‘𝐴) = (((𝑄‘𝑀) ∘f + (𝑄‘𝑁))‘𝐴)) |
33 | 8, 6, 23, 4, 24, 27 | pwselbas 17434 | . . . . 5 ⊢ (𝜑 → (𝑄‘𝑀):(𝐾 ↑m 𝐼)⟶𝐾) |
34 | 33 | ffnd 6718 | . . . 4 ⊢ (𝜑 → (𝑄‘𝑀) Fn (𝐾 ↑m 𝐼)) |
35 | 8, 6, 23, 4, 24, 28 | pwselbas 17434 | . . . . 5 ⊢ (𝜑 → (𝑄‘𝑁):(𝐾 ↑m 𝐼)⟶𝐾) |
36 | 35 | ffnd 6718 | . . . 4 ⊢ (𝜑 → (𝑄‘𝑁) Fn (𝐾 ↑m 𝐼)) |
37 | evladdval.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) | |
38 | fnfvof 7686 | . . . 4 ⊢ ((((𝑄‘𝑀) Fn (𝐾 ↑m 𝐼) ∧ (𝑄‘𝑁) Fn (𝐾 ↑m 𝐼)) ∧ ((𝐾 ↑m 𝐼) ∈ V ∧ 𝐴 ∈ (𝐾 ↑m 𝐼))) → (((𝑄‘𝑀) ∘f + (𝑄‘𝑁))‘𝐴) = (((𝑄‘𝑀)‘𝐴) + ((𝑄‘𝑁)‘𝐴))) | |
39 | 34, 36, 24, 37, 38 | syl22anc 837 | . . 3 ⊢ (𝜑 → (((𝑄‘𝑀) ∘f + (𝑄‘𝑁))‘𝐴) = (((𝑄‘𝑀)‘𝐴) + ((𝑄‘𝑁)‘𝐴))) |
40 | 15 | simprd 496 | . . . 4 ⊢ (𝜑 → ((𝑄‘𝑀)‘𝐴) = 𝑉) |
41 | 17 | simprd 496 | . . . 4 ⊢ (𝜑 → ((𝑄‘𝑁)‘𝐴) = 𝑊) |
42 | 40, 41 | oveq12d 7426 | . . 3 ⊢ (𝜑 → (((𝑄‘𝑀)‘𝐴) + ((𝑄‘𝑁)‘𝐴)) = (𝑉 + 𝑊)) |
43 | 32, 39, 42 | 3eqtrd 2776 | . 2 ⊢ (𝜑 → ((𝑄‘(𝑀 ✚ 𝑁))‘𝐴) = (𝑉 + 𝑊)) |
44 | 19, 43 | jca 512 | 1 ⊢ (𝜑 → ((𝑀 ✚ 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 ✚ 𝑁))‘𝐴) = (𝑉 + 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ∘f cof 7667 ↑m cmap 8819 Basecbs 17143 +gcplusg 17196 ↑s cpws 17391 Grpcgrp 18818 GrpHom cghm 19088 CRingccrg 20056 RingHom crh 20247 mPoly cmpl 21458 eval cevl 21633 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-ofr 7670 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-fzo 13627 df-seq 13966 df-hash 14290 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mulg 18950 df-subg 19002 df-ghm 19089 df-cntz 19180 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-srg 20009 df-ring 20057 df-cring 20058 df-rnghom 20250 df-subrg 20316 df-lmod 20472 df-lss 20542 df-lsp 20582 df-assa 21407 df-asp 21408 df-ascl 21409 df-psr 21461 df-mvr 21462 df-mpl 21463 df-evls 21634 df-evl 21635 |
This theorem is referenced by: selvadd 41162 |
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