evl1muld - Metamath Proof Explorer

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Theorem evl1muld 22263

Description: Polynomial evaluation builder for multiplication 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))
evl1muld.t	⊢ ∙ = (._r‘𝑃)
evl1muld.s	⊢ · = (._r‘𝑅)

**Assertion**
Ref	Expression
evl1muld	⊢ (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ∙ 𝑁))‘𝑌) = (𝑉 · 𝑊)))

**Proof of Theorem evl1muld**
Step	Hyp	Ref	Expression
1		evl1addd.1	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
2		evl1addd.q	. . . . . 6 ⊢ 𝑂 = (eval₁‘𝑅)
3		evl1addd.p	. . . . . 6 ⊢ 𝑃 = (Poly₁‘𝑅)
4		eqid 2729	. . . . . 6 ⊢ (𝑅 ↑_s 𝐵) = (𝑅 ↑_s 𝐵)
5		evl1addd.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
6	2, 3, 4, 5	evl1rhm 22252	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)))
7	1, 6	syl 17	. . . 4 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)))
8		rhmrcl1 20396	. . . 4 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)) → 𝑃 ∈ Ring)
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → 𝑃 ∈ Ring)
10		evl1addd.3	. . . 4 ⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉))
11	10	simpld 494	. . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
12		evl1addd.4	. . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊))
13	12	simpld 494	. . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑈)
14		evl1addd.u	. . . 4 ⊢ 𝑈 = (Base‘𝑃)
15		evl1muld.t	. . . 4 ⊢ ∙ = (._r‘𝑃)
16	14, 15	ringcl 20170	. . 3 ⊢ ((𝑃 ∈ Ring ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑀 ∙ 𝑁) ∈ 𝑈)
17	9, 11, 13, 16	syl3anc 1373	. 2 ⊢ (𝜑 → (𝑀 ∙ 𝑁) ∈ 𝑈)
18		eqid 2729	. . . . . . 7 ⊢ (._r‘(𝑅 ↑_s 𝐵)) = (._r‘(𝑅 ↑_s 𝐵))
19	14, 15, 18	rhmmul 20406	. . . . . 6 ⊢ ((𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)) ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑂‘(𝑀 ∙ 𝑁)) = ((𝑂‘𝑀)(._r‘(𝑅 ↑_s 𝐵))(𝑂‘𝑁)))
20	7, 11, 13, 19	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 ∙ 𝑁)) = ((𝑂‘𝑀)(._r‘(𝑅 ↑_s 𝐵))(𝑂‘𝑁)))
21		eqid 2729	. . . . . 6 ⊢ (Base‘(𝑅 ↑_s 𝐵)) = (Base‘(𝑅 ↑_s 𝐵))
22	5	fvexi 6854	. . . . . . 7 ⊢ 𝐵 ∈ V
23	22	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
24	14, 21	rhmf 20405	. . . . . . . 8 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)) → 𝑂:𝑈⟶(Base‘(𝑅 ↑_s 𝐵)))
25	7, 24	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑂:𝑈⟶(Base‘(𝑅 ↑_s 𝐵)))
26	25, 11	ffvelcdmd 7039	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑_s 𝐵)))
27	25, 13	ffvelcdmd 7039	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑁) ∈ (Base‘(𝑅 ↑_s 𝐵)))
28		evl1muld.s	. . . . . 6 ⊢ · = (._r‘𝑅)
29	4, 21, 1, 23, 26, 27, 28, 18	pwsmulrval 17430	. . . . 5 ⊢ (𝜑 → ((𝑂‘𝑀)(._r‘(𝑅 ↑_s 𝐵))(𝑂‘𝑁)) = ((𝑂‘𝑀) ∘_f · (𝑂‘𝑁)))
30	20, 29	eqtrd 2764	. . . 4 ⊢ (𝜑 → (𝑂‘(𝑀 ∙ 𝑁)) = ((𝑂‘𝑀) ∘_f · (𝑂‘𝑁)))
31	30	fveq1d 6842	. . 3 ⊢ (𝜑 → ((𝑂‘(𝑀 ∙ 𝑁))‘𝑌) = (((𝑂‘𝑀) ∘_f · (𝑂‘𝑁))‘𝑌))
32	4, 5, 21, 1, 23, 26	pwselbas 17428	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵)
33	32	ffnd 6671	. . . 4 ⊢ (𝜑 → (𝑂‘𝑀) Fn 𝐵)
34	4, 5, 21, 1, 23, 27	pwselbas 17428	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑁):𝐵⟶𝐵)
35	34	ffnd 6671	. . . 4 ⊢ (𝜑 → (𝑂‘𝑁) Fn 𝐵)
36		evl1addd.2	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
37		fnfvof 7650	. . . 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)))
38	33, 35, 23, 36, 37	syl22anc 838	. . 3 ⊢ (𝜑 → (((𝑂‘𝑀) ∘_f · (𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌) · ((𝑂‘𝑁)‘𝑌)))
39	10	simprd 495	. . . 4 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) = 𝑉)
40	12	simprd 495	. . . 4 ⊢ (𝜑 → ((𝑂‘𝑁)‘𝑌) = 𝑊)
41	39, 40	oveq12d 7387	. . 3 ⊢ (𝜑 → (((𝑂‘𝑀)‘𝑌) · ((𝑂‘𝑁)‘𝑌)) = (𝑉 · 𝑊))
42	31, 38, 41	3eqtrd 2768	. 2 ⊢ (𝜑 → ((𝑂‘(𝑀 ∙ 𝑁))‘𝑌) = (𝑉 · 𝑊))
43	17, 42	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 1540 ∈ wcel 2109 Vcvv 3444 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ∘_f cof 7631 Basecbs 17155 ._rcmulr 17197 ↑_s cpws 17385 Ringcrg 20153 CRingccrg 20154 RingHom crh 20389 Poly₁cpl1 22094 eval₁ce1 22234

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-ofr 7634 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-sup 9369 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-fz 13445 df-fzo 13592 df-seq 13943 df-hash 14272 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 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 17380 df-gsum 17381 df-prds 17386 df-pws 17388 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-mhm 18692 df-submnd 18693 df-grp 18850 df-minusg 18851 df-sbg 18852 df-mulg 18982 df-subg 19037 df-ghm 19127 df-cntz 19231 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-srg 20107 df-ring 20155 df-cring 20156 df-rhm 20392 df-subrng 20466 df-subrg 20490 df-lmod 20800 df-lss 20870 df-lsp 20910 df-assa 21795 df-asp 21796 df-ascl 21797 df-psr 21851 df-mvr 21852 df-mpl 21853 df-opsr 21855 df-evls 22014 df-evl 22015 df-psr1 22097 df-ply1 22099 df-evl1 22236

This theorem is referenced by: evl1vsd 22264 evls1muld 22292 ply1mulrtss 33543 aks6d1c1p4 42092 evl1gprodd 42098 aks6d1c5lem2 42119

W3C validator