evl1addd - Metamath Proof Explorer

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Theorem evl1addd 20027

Description: Polynomial evaluation builder for addition of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
evl1addd	⊢ (𝜑 → ((𝑀 ✚ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊)))

**Proof of Theorem evl1addd**
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 2799	. . . . . . 7 ⊢ (𝑅 ↑_s 𝐵) = (𝑅 ↑_s 𝐵)
5		evl1addd.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
6	2, 3, 4, 5	evl1rhm 20018	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)))
7	1, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)))
8		rhmghm 19043	. . . . 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 17975	. . . 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 489	. . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
14		evl1addd.4	. . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊))
15	14	simpld 489	. . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑈)
16		evl1addd.u	. . . 4 ⊢ 𝑈 = (Base‘𝑃)
17		evl1addd.g	. . . 4 ⊢ ✚ = (+_g‘𝑃)
18	16, 17	grpcl 17746	. . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑀 ✚ 𝑁) ∈ 𝑈)
19	11, 13, 15, 18	syl3anc 1491	. 2 ⊢ (𝜑 → (𝑀 ✚ 𝑁) ∈ 𝑈)
20		eqid 2799	. . . . . . 7 ⊢ (+_g‘(𝑅 ↑_s 𝐵)) = (+_g‘(𝑅 ↑_s 𝐵))
21	16, 17, 20	ghmlin 17978	. . . . . 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 1491	. . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 ✚ 𝑁)) = ((𝑂‘𝑀)(+_g‘(𝑅 ↑_s 𝐵))(𝑂‘𝑁)))
23		eqid 2799	. . . . . 6 ⊢ (Base‘(𝑅 ↑_s 𝐵)) = (Base‘(𝑅 ↑_s 𝐵))
24	5	fvexi 6425	. . . . . . 7 ⊢ 𝐵 ∈ V
25	24	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
26	16, 23	rhmf 19044	. . . . . . . 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 6586	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑_s 𝐵)))
29	27, 15	ffvelrnd 6586	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑁) ∈ (Base‘(𝑅 ↑_s 𝐵)))
30		evl1addd.a	. . . . . 6 ⊢ + = (+_g‘𝑅)
31	4, 23, 1, 25, 28, 29, 30, 20	pwsplusgval 16465	. . . . 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 6413	. . 3 ⊢ (𝜑 → ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (((𝑂‘𝑀) ∘_𝑓 + (𝑂‘𝑁))‘𝑌))
34	4, 5, 23, 1, 25, 28	pwselbas 16464	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵)
35	34	ffnd 6257	. . . 4 ⊢ (𝜑 → (𝑂‘𝑀) Fn 𝐵)
36	4, 5, 23, 1, 25, 29	pwselbas 16464	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑁):𝐵⟶𝐵)
37	36	ffnd 6257	. . . 4 ⊢ (𝜑 → (𝑂‘𝑁) Fn 𝐵)
38		evl1addd.2	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
39		fnfvof 7145	. . . 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 868	. . 3 ⊢ (𝜑 → (((𝑂‘𝑀) ∘_𝑓 + (𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌) + ((𝑂‘𝑁)‘𝑌)))
41	12	simprd 490	. . . 4 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) = 𝑉)
42	14	simprd 490	. . . 4 ⊢ (𝜑 → ((𝑂‘𝑁)‘𝑌) = 𝑊)
43	41, 42	oveq12d 6896	. . 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 508	1 ⊢ (𝜑 → ((𝑀 ✚ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 Vcvv 3385 Fn wfn 6096 ⟶wf 6097 ‘cfv 6101 (class class class)co 6878 ∘_𝑓 cof 7129 Basecbs 16184 +_gcplusg 16267 ↑_s cpws 16422 Grpcgrp 17738 GrpHom cghm 17970 CRingccrg 18864 RingHom crh 19030 Poly₁cpl1 19869 eval₁ce1 20001

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-ofr 7132 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-sup 8590 df-oi 8657 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-fz 12581 df-fzo 12721 df-seq 13056 df-hash 13371 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-sca 16283 df-vsca 16284 df-ip 16285 df-tset 16286 df-ple 16287 df-ds 16289 df-hom 16291 df-cco 16292 df-0g 16417 df-gsum 16418 df-prds 16423 df-pws 16425 df-mre 16561 df-mrc 16562 df-acs 16564 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-mhm 17650 df-submnd 17651 df-grp 17741 df-minusg 17742 df-sbg 17743 df-mulg 17857 df-subg 17904 df-ghm 17971 df-cntz 18062 df-cmn 18510 df-abl 18511 df-mgp 18806 df-ur 18818 df-srg 18822 df-ring 18865 df-cring 18866 df-rnghom 19033 df-subrg 19096 df-lmod 19183 df-lss 19251 df-lsp 19293 df-assa 19635 df-asp 19636 df-ascl 19637 df-psr 19679 df-mvr 19680 df-mpl 19681 df-opsr 19683 df-evls 19828 df-evl 19829 df-psr1 19872 df-ply1 19874 df-evl1 20003

This theorem is referenced by: evl1gsumdlem 20042

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