evl1addd - Metamath Proof Explorer

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Theorem evl1addd 22267

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 2725	. . . . . . 7 ⊢ (𝑅 ↑_s 𝐵) = (𝑅 ↑_s 𝐵)
5		evl1addd.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
6	2, 3, 4, 5	evl1rhm 22258	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)))
7	1, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)))
8		rhmghm 20425	. . . . 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 19174	. . . 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 493	. . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
14		evl1addd.4	. . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊))
15	14	simpld 493	. . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑈)
16		evl1addd.u	. . . 4 ⊢ 𝑈 = (Base‘𝑃)
17		evl1addd.g	. . . 4 ⊢ ✚ = (+_g‘𝑃)
18	16, 17	grpcl 18900	. . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑀 ✚ 𝑁) ∈ 𝑈)
19	11, 13, 15, 18	syl3anc 1368	. 2 ⊢ (𝜑 → (𝑀 ✚ 𝑁) ∈ 𝑈)
20		eqid 2725	. . . . . . 7 ⊢ (+_g‘(𝑅 ↑_s 𝐵)) = (+_g‘(𝑅 ↑_s 𝐵))
21	16, 17, 20	ghmlin 19177	. . . . . 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 2725	. . . . . 6 ⊢ (Base‘(𝑅 ↑_s 𝐵)) = (Base‘(𝑅 ↑_s 𝐵))
24	5	fvexi 6904	. . . . . . 7 ⊢ 𝐵 ∈ V
25	24	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
26	16, 23	rhmf 20426	. . . . . . . 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 7088	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑_s 𝐵)))
29	27, 15	ffvelcdmd 7088	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑁) ∈ (Base‘(𝑅 ↑_s 𝐵)))
30		evl1addd.a	. . . . . 6 ⊢ + = (+_g‘𝑅)
31	4, 23, 1, 25, 28, 29, 30, 20	pwsplusgval 17469	. . . . 5 ⊢ (𝜑 → ((𝑂‘𝑀)(+_g‘(𝑅 ↑_s 𝐵))(𝑂‘𝑁)) = ((𝑂‘𝑀) ∘_f + (𝑂‘𝑁)))
32	22, 31	eqtrd 2765	. . . 4 ⊢ (𝜑 → (𝑂‘(𝑀 ✚ 𝑁)) = ((𝑂‘𝑀) ∘_f + (𝑂‘𝑁)))
33	32	fveq1d 6892	. . 3 ⊢ (𝜑 → ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (((𝑂‘𝑀) ∘_f + (𝑂‘𝑁))‘𝑌))
34	4, 5, 23, 1, 25, 28	pwselbas 17468	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵)
35	34	ffnd 6716	. . . 4 ⊢ (𝜑 → (𝑂‘𝑀) Fn 𝐵)
36	4, 5, 23, 1, 25, 29	pwselbas 17468	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑁):𝐵⟶𝐵)
37	36	ffnd 6716	. . . 4 ⊢ (𝜑 → (𝑂‘𝑁) Fn 𝐵)
38		evl1addd.2	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
39		fnfvof 7697	. . . 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 494	. . . 4 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) = 𝑉)
42	14	simprd 494	. . . 4 ⊢ (𝜑 → ((𝑂‘𝑁)‘𝑌) = 𝑊)
43	41, 42	oveq12d 7432	. . 3 ⊢ (𝜑 → (((𝑂‘𝑀)‘𝑌) + ((𝑂‘𝑁)‘𝑌)) = (𝑉 + 𝑊))
44	33, 40, 43	3eqtrd 2769	. 2 ⊢ (𝜑 → ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊))
45	19, 44	jca 510	1 ⊢ (𝜑 → ((𝑀 ✚ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7414 ∘_f cof 7678 Basecbs 17177 +_gcplusg 17230 ↑_s cpws 17425 Grpcgrp 18892 GrpHom cghm 19169 CRingccrg 20176 RingHom crh 20410 Poly₁cpl1 22102 eval₁ce1 22240

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-ofr 7681 df-om 7867 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-sup 9463 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-fzo 13658 df-seq 13997 df-hash 14320 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-hom 17254 df-cco 17255 df-0g 17420 df-gsum 17421 df-prds 17426 df-pws 17428 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-ghm 19170 df-cntz 19270 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-srg 20129 df-ring 20177 df-cring 20178 df-rhm 20413 df-subrng 20485 df-subrg 20510 df-lmod 20747 df-lss 20818 df-lsp 20858 df-assa 21789 df-asp 21790 df-ascl 21791 df-psr 21844 df-mvr 21845 df-mpl 21846 df-opsr 21848 df-evls 22023 df-evl 22024 df-psr1 22105 df-ply1 22107 df-evl1 22242

This theorem is referenced by: evl1gsumdlem 22282 evls1addd 22297 aks6d1c1p2 41608 aks6d1c1p3 41609 aks6d1c5lem1 41635 aks6d1c5lem2 41637

W3C validator