evl1addd - Metamath Proof Explorer

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Theorem evl1addd 22256

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 2731	. . . . . . 7 ⊢ (𝑅 ↑_s 𝐵) = (𝑅 ↑_s 𝐵)
5		evl1addd.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
6	2, 3, 4, 5	evl1rhm 22247	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)))
7	1, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)))
8		rhmghm 20401	. . . . 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 19130	. . . 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 18854	. . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑀 ✚ 𝑁) ∈ 𝑈)
19	11, 13, 15, 18	syl3anc 1373	. 2 ⊢ (𝜑 → (𝑀 ✚ 𝑁) ∈ 𝑈)
20		eqid 2731	. . . . . . 7 ⊢ (+_g‘(𝑅 ↑_s 𝐵)) = (+_g‘(𝑅 ↑_s 𝐵))
21	16, 17, 20	ghmlin 19133	. . . . . 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 2731	. . . . . 6 ⊢ (Base‘(𝑅 ↑_s 𝐵)) = (Base‘(𝑅 ↑_s 𝐵))
24	5	fvexi 6836	. . . . . . 7 ⊢ 𝐵 ∈ V
25	24	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
26	16, 23	rhmf 20402	. . . . . . . 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 7018	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑_s 𝐵)))
29	27, 15	ffvelcdmd 7018	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑁) ∈ (Base‘(𝑅 ↑_s 𝐵)))
30		evl1addd.a	. . . . . 6 ⊢ + = (+_g‘𝑅)
31	4, 23, 1, 25, 28, 29, 30, 20	pwsplusgval 17394	. . . . 5 ⊢ (𝜑 → ((𝑂‘𝑀)(+_g‘(𝑅 ↑_s 𝐵))(𝑂‘𝑁)) = ((𝑂‘𝑀) ∘_f + (𝑂‘𝑁)))
32	22, 31	eqtrd 2766	. . . 4 ⊢ (𝜑 → (𝑂‘(𝑀 ✚ 𝑁)) = ((𝑂‘𝑀) ∘_f + (𝑂‘𝑁)))
33	32	fveq1d 6824	. . 3 ⊢ (𝜑 → ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (((𝑂‘𝑀) ∘_f + (𝑂‘𝑁))‘𝑌))
34	4, 5, 23, 1, 25, 28	pwselbas 17393	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵)
35	34	ffnd 6652	. . . 4 ⊢ (𝜑 → (𝑂‘𝑀) Fn 𝐵)
36	4, 5, 23, 1, 25, 29	pwselbas 17393	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑁):𝐵⟶𝐵)
37	36	ffnd 6652	. . . 4 ⊢ (𝜑 → (𝑂‘𝑁) Fn 𝐵)
38		evl1addd.2	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
39		fnfvof 7627	. . . 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 838	. . 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 7364	. . 3 ⊢ (𝜑 → (((𝑂‘𝑀)‘𝑌) + ((𝑂‘𝑁)‘𝑌)) = (𝑉 + 𝑊))
44	33, 40, 43	3eqtrd 2770	. 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 1541 ∈ wcel 2111 Vcvv 3436 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ∘_f cof 7608 Basecbs 17120 +_gcplusg 17161 ↑_s cpws 17350 Grpcgrp 18846 GrpHom cghm 19124 CRingccrg 20152 RingHom crh 20387 Poly₁cpl1 22089 eval₁ce1 22229

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-ofr 7611 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-fz 13408 df-fzo 13555 df-seq 13909 df-hash 14238 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-ghm 19125 df-cntz 19229 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-srg 20105 df-ring 20153 df-cring 20154 df-rhm 20390 df-subrng 20461 df-subrg 20485 df-lmod 20795 df-lss 20865 df-lsp 20905 df-assa 21790 df-asp 21791 df-ascl 21792 df-psr 21846 df-mvr 21847 df-mpl 21848 df-opsr 21850 df-evls 22009 df-evl 22010 df-psr1 22092 df-ply1 22094 df-evl1 22231

This theorem is referenced by: evl1gsumdlem 22271 evls1addd 22286 aks6d1c1p2 42212 aks6d1c1p3 42213 aks6d1c5lem1 42239 aks6d1c5lem2 42241 aks5lem3a 42292

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