evlsmulval - Metamath Proof Explorer

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Theorem evlsmulval 22180

Description: Polynomial evaluation builder for multiplication. (Contributed by SN, 27-Jul-2024.)

**Hypotheses**
Ref	Expression
evlsaddval.q	⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
evlsaddval.p	⊢ 𝑃 = (𝐼 mPoly 𝑈)
evlsaddval.u	⊢ 𝑈 = (𝑆 ↾_s 𝑅)
evlsaddval.k	⊢ 𝐾 = (Base‘𝑆)
evlsaddval.b	⊢ 𝐵 = (Base‘𝑃)
evlsaddval.i	⊢ (𝜑 → 𝐼 ∈ 𝑍)
evlsaddval.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evlsaddval.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
evlsaddval.a	⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
evlsaddval.m	⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉))
evlsaddval.n	⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ ((𝑄‘𝑁)‘𝐴) = 𝑊))
evlsmulval.g	⊢ ∙ = (._r‘𝑃)
evlsmulval.f	⊢ · = (._r‘𝑆)

**Assertion**
Ref	Expression
evlsmulval	⊢ (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (𝑉 · 𝑊)))

**Proof of Theorem evlsmulval**
Step	Hyp	Ref	Expression
1		evlsaddval.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑍)
2		evlsaddval.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ CRing)
3		evlsaddval.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
4		evlsaddval.q	. . . . . 6 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
5		evlsaddval.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑈)
6		evlsaddval.u	. . . . . 6 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
7		eqid 2762	. . . . . 6 ⊢ (𝑆 ↑_s (𝐾 ↑_m 𝐼)) = (𝑆 ↑_s (𝐾 ↑_m 𝐼))
8		evlsaddval.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑆)
9	4, 5, 6, 7, 8	evlsrhm 22138	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))))
10	1, 2, 3, 9	syl3anc 1390	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))))
11		rhmrcl1 20521	. . . 4 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))) → 𝑃 ∈ Ring)
12	10, 11	syl 17	. . 3 ⊢ (𝜑 → 𝑃 ∈ Ring)
13		evlsaddval.m	. . . 4 ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉))
14	13	simpld 498	. . 3 ⊢ (𝜑 → 𝑀 ∈ 𝐵)
15		evlsaddval.n	. . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ ((𝑄‘𝑁)‘𝐴) = 𝑊))
16	15	simpld 498	. . 3 ⊢ (𝜑 → 𝑁 ∈ 𝐵)
17		evlsaddval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
18		evlsmulval.g	. . . 4 ⊢ ∙ = (._r‘𝑃)
19	17, 18	ringcl 20296	. . 3 ⊢ ((𝑃 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵) → (𝑀 ∙ 𝑁) ∈ 𝐵)
20	12, 14, 16, 19	syl3anc 1390	. 2 ⊢ (𝜑 → (𝑀 ∙ 𝑁) ∈ 𝐵)
21		eqid 2762	. . . . . . 7 ⊢ (._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
22	17, 18, 21	rhmmul 20531	. . . . . 6 ⊢ ((𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∧ 𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵) → (𝑄‘(𝑀 ∙ 𝑁)) = ((𝑄‘𝑀)(._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))(𝑄‘𝑁)))
23	10, 14, 16, 22	syl3anc 1390	. . . . 5 ⊢ (𝜑 → (𝑄‘(𝑀 ∙ 𝑁)) = ((𝑄‘𝑀)(._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))(𝑄‘𝑁)))
24		eqid 2762	. . . . . 6 ⊢ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
25		ovexd 7431	. . . . . 6 ⊢ (𝜑 → (𝐾 ↑_m 𝐼) ∈ V)
26	17, 24	rhmf 20529	. . . . . . . 8 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))) → 𝑄:𝐵⟶(Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
27	10, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑄:𝐵⟶(Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
28	27, 14	ffvelcdmd 7066	. . . . . 6 ⊢ (𝜑 → (𝑄‘𝑀) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
29	27, 16	ffvelcdmd 7066	. . . . . 6 ⊢ (𝜑 → (𝑄‘𝑁) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
30		evlsmulval.f	. . . . . 6 ⊢ · = (._r‘𝑆)
31	7, 24, 2, 25, 28, 29, 30, 21	pwsmulrval 17521	. . . . 5 ⊢ (𝜑 → ((𝑄‘𝑀)(._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))(𝑄‘𝑁)) = ((𝑄‘𝑀) ∘_f · (𝑄‘𝑁)))
32	23, 31	eqtrd 2797	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑀 ∙ 𝑁)) = ((𝑄‘𝑀) ∘_f · (𝑄‘𝑁)))
33	32	fveq1d 6869	. . 3 ⊢ (𝜑 → ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (((𝑄‘𝑀) ∘_f · (𝑄‘𝑁))‘𝐴))
34	7, 8, 24, 2, 25, 28	pwselbas 17518	. . . . 5 ⊢ (𝜑 → (𝑄‘𝑀):(𝐾 ↑_m 𝐼)⟶𝐾)
35	34	ffnd 6692	. . . 4 ⊢ (𝜑 → (𝑄‘𝑀) Fn (𝐾 ↑_m 𝐼))
36	7, 8, 24, 2, 25, 29	pwselbas 17518	. . . . 5 ⊢ (𝜑 → (𝑄‘𝑁):(𝐾 ↑_m 𝐼)⟶𝐾)
37	36	ffnd 6692	. . . 4 ⊢ (𝜑 → (𝑄‘𝑁) Fn (𝐾 ↑_m 𝐼))
38		evlsaddval.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
39		fnfvof 7677	. . . 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)))
40	35, 37, 25, 38, 39	syl22anc 849	. . 3 ⊢ (𝜑 → (((𝑄‘𝑀) ∘_f · (𝑄‘𝑁))‘𝐴) = (((𝑄‘𝑀)‘𝐴) · ((𝑄‘𝑁)‘𝐴)))
41	13	simprd 499	. . . 4 ⊢ (𝜑 → ((𝑄‘𝑀)‘𝐴) = 𝑉)
42	15	simprd 499	. . . 4 ⊢ (𝜑 → ((𝑄‘𝑁)‘𝐴) = 𝑊)
43	41, 42	oveq12d 7414	. . 3 ⊢ (𝜑 → (((𝑄‘𝑀)‘𝐴) · ((𝑄‘𝑁)‘𝐴)) = (𝑉 · 𝑊))
44	33, 40, 43	3eqtrd 2801	. 2 ⊢ (𝜑 → ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (𝑉 · 𝑊))
45	20, 44	jca 519	1 ⊢ (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (𝑉 · 𝑊)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1560 ∈ wcel 2142 Vcvv 3454 Fn wfn 6516 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 ∘_f cof 7658 ↑_m cmap 8808 Basecbs 17245 ↾_s cress 17266 ._rcmulr 17287 ↑_s cpws 17475 Ringcrg 20279 CRingccrg 20280 RingHom crh 20514 SubRingcsubrg 20615 mPoly cmpl 21955 evalSub ces 22122

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-ofr 7661 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-sup 9388 df-oi 9458 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-fz 13513 df-fzo 13660 df-seq 14015 df-hash 14344 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-hom 17310 df-cco 17311 df-0g 17470 df-gsum 17471 df-prds 17476 df-pws 17478 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-mhm 18817 df-submnd 18818 df-grp 18978 df-minusg 18979 df-sbg 18980 df-mulg 19110 df-subg 19165 df-ghm 19254 df-cntz 19357 df-cmn 19822 df-abl 19823 df-mgp 20187 df-rng 20199 df-ur 20228 df-srg 20233 df-ring 20281 df-cring 20282 df-rhm 20517 df-subrng 20592 df-subrg 20616 df-lmod 20926 df-lss 20996 df-lsp 21036 df-assa 21902 df-asp 21903 df-ascl 21904 df-psr 21958 df-mvr 21959 df-mpl 21960 df-evls 22124

This theorem is referenced by: evlsmaprhm 22181

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