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Mirrors > Home > MPE Home > Th. List > evls1muld | Structured version Visualization version GIF version |
Description: Univariate polynomial evaluation of a product of polynomials. (Contributed by Thierry Arnoux, 24-Jan-2025.) |
Ref | Expression |
---|---|
ressply1evl2.q | ⊢ 𝑄 = (𝑆 evalSub1 𝑅) |
ressply1evl2.k | ⊢ 𝐾 = (Base‘𝑆) |
ressply1evl2.w | ⊢ 𝑊 = (Poly1‘𝑈) |
ressply1evl2.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
ressply1evl2.b | ⊢ 𝐵 = (Base‘𝑊) |
evls1muld.1 | ⊢ × = (.r‘𝑊) |
evls1muld.2 | ⊢ · = (.r‘𝑆) |
evls1muld.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evls1muld.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evls1muld.m | ⊢ (𝜑 → 𝑀 ∈ 𝐵) |
evls1muld.n | ⊢ (𝜑 → 𝑁 ∈ 𝐵) |
evls1muld.c | ⊢ (𝜑 → 𝐶 ∈ 𝐾) |
Ref | Expression |
---|---|
evls1muld | ⊢ (𝜑 → ((𝑄‘(𝑀 × 𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . . 6 ⊢ (𝜑 → 𝜑) | |
2 | evls1muld.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝐵) | |
3 | evls1muld.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ 𝐵) | |
4 | eqid 2735 | . . . . . . 7 ⊢ (Poly1‘𝑆) = (Poly1‘𝑆) | |
5 | ressply1evl2.u | . . . . . . 7 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
6 | ressply1evl2.w | . . . . . . 7 ⊢ 𝑊 = (Poly1‘𝑈) | |
7 | ressply1evl2.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑊) | |
8 | evls1muld.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
9 | eqid 2735 | . . . . . . 7 ⊢ ((Poly1‘𝑆) ↾s 𝐵) = ((Poly1‘𝑆) ↾s 𝐵) | |
10 | 4, 5, 6, 7, 8, 9 | ressply1mul 22248 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵)) → (𝑀(.r‘𝑊)𝑁) = (𝑀(.r‘((Poly1‘𝑆) ↾s 𝐵))𝑁)) |
11 | 1, 2, 3, 10 | syl12anc 837 | . . . . 5 ⊢ (𝜑 → (𝑀(.r‘𝑊)𝑁) = (𝑀(.r‘((Poly1‘𝑆) ↾s 𝐵))𝑁)) |
12 | evls1muld.1 | . . . . . 6 ⊢ × = (.r‘𝑊) | |
13 | 12 | oveqi 7444 | . . . . 5 ⊢ (𝑀 × 𝑁) = (𝑀(.r‘𝑊)𝑁) |
14 | 7 | fvexi 6921 | . . . . . . 7 ⊢ 𝐵 ∈ V |
15 | eqid 2735 | . . . . . . . 8 ⊢ (.r‘(Poly1‘𝑆)) = (.r‘(Poly1‘𝑆)) | |
16 | 9, 15 | ressmulr 17353 | . . . . . . 7 ⊢ (𝐵 ∈ V → (.r‘(Poly1‘𝑆)) = (.r‘((Poly1‘𝑆) ↾s 𝐵))) |
17 | 14, 16 | ax-mp 5 | . . . . . 6 ⊢ (.r‘(Poly1‘𝑆)) = (.r‘((Poly1‘𝑆) ↾s 𝐵)) |
18 | 17 | oveqi 7444 | . . . . 5 ⊢ (𝑀(.r‘(Poly1‘𝑆))𝑁) = (𝑀(.r‘((Poly1‘𝑆) ↾s 𝐵))𝑁) |
19 | 11, 13, 18 | 3eqtr4g 2800 | . . . 4 ⊢ (𝜑 → (𝑀 × 𝑁) = (𝑀(.r‘(Poly1‘𝑆))𝑁)) |
20 | 19 | fveq2d 6911 | . . 3 ⊢ (𝜑 → ((eval1‘𝑆)‘(𝑀 × 𝑁)) = ((eval1‘𝑆)‘(𝑀(.r‘(Poly1‘𝑆))𝑁))) |
21 | 20 | fveq1d 6909 | . 2 ⊢ (𝜑 → (((eval1‘𝑆)‘(𝑀 × 𝑁))‘𝐶) = (((eval1‘𝑆)‘(𝑀(.r‘(Poly1‘𝑆))𝑁))‘𝐶)) |
22 | ressply1evl2.q | . . . . . 6 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
23 | ressply1evl2.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
24 | eqid 2735 | . . . . . 6 ⊢ (eval1‘𝑆) = (eval1‘𝑆) | |
25 | evls1muld.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
26 | 22, 23, 6, 5, 7, 24, 25, 8 | ressply1evl 22390 | . . . . 5 ⊢ (𝜑 → 𝑄 = ((eval1‘𝑆) ↾ 𝐵)) |
27 | 26 | fveq1d 6909 | . . . 4 ⊢ (𝜑 → (𝑄‘(𝑀 × 𝑁)) = (((eval1‘𝑆) ↾ 𝐵)‘(𝑀 × 𝑁))) |
28 | 5 | subrgring 20591 | . . . . . . 7 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring) |
29 | 6 | ply1ring 22265 | . . . . . . 7 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring) |
30 | 8, 28, 29 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Ring) |
31 | 7, 12, 30, 2, 3 | ringcld 20277 | . . . . 5 ⊢ (𝜑 → (𝑀 × 𝑁) ∈ 𝐵) |
32 | 31 | fvresd 6927 | . . . 4 ⊢ (𝜑 → (((eval1‘𝑆) ↾ 𝐵)‘(𝑀 × 𝑁)) = ((eval1‘𝑆)‘(𝑀 × 𝑁))) |
33 | 27, 32 | eqtr2d 2776 | . . 3 ⊢ (𝜑 → ((eval1‘𝑆)‘(𝑀 × 𝑁)) = (𝑄‘(𝑀 × 𝑁))) |
34 | 33 | fveq1d 6909 | . 2 ⊢ (𝜑 → (((eval1‘𝑆)‘(𝑀 × 𝑁))‘𝐶) = ((𝑄‘(𝑀 × 𝑁))‘𝐶)) |
35 | eqid 2735 | . . . 4 ⊢ (Base‘(Poly1‘𝑆)) = (Base‘(Poly1‘𝑆)) | |
36 | evls1muld.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
37 | eqid 2735 | . . . . . . . 8 ⊢ (PwSer1‘𝑈) = (PwSer1‘𝑈) | |
38 | eqid 2735 | . . . . . . . 8 ⊢ (Base‘(PwSer1‘𝑈)) = (Base‘(PwSer1‘𝑈)) | |
39 | 4, 5, 6, 7, 8, 37, 38, 35 | ressply1bas2 22245 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = ((Base‘(PwSer1‘𝑈)) ∩ (Base‘(Poly1‘𝑆)))) |
40 | inss2 4246 | . . . . . . 7 ⊢ ((Base‘(PwSer1‘𝑈)) ∩ (Base‘(Poly1‘𝑆))) ⊆ (Base‘(Poly1‘𝑆)) | |
41 | 39, 40 | eqsstrdi 4050 | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(Poly1‘𝑆))) |
42 | 41, 2 | sseldd 3996 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (Base‘(Poly1‘𝑆))) |
43 | 26 | fveq1d 6909 | . . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = (((eval1‘𝑆) ↾ 𝐵)‘𝑀)) |
44 | 2 | fvresd 6927 | . . . . . . 7 ⊢ (𝜑 → (((eval1‘𝑆) ↾ 𝐵)‘𝑀) = ((eval1‘𝑆)‘𝑀)) |
45 | 43, 44 | eqtr2d 2776 | . . . . . 6 ⊢ (𝜑 → ((eval1‘𝑆)‘𝑀) = (𝑄‘𝑀)) |
46 | 45 | fveq1d 6909 | . . . . 5 ⊢ (𝜑 → (((eval1‘𝑆)‘𝑀)‘𝐶) = ((𝑄‘𝑀)‘𝐶)) |
47 | 42, 46 | jca 511 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ (Base‘(Poly1‘𝑆)) ∧ (((eval1‘𝑆)‘𝑀)‘𝐶) = ((𝑄‘𝑀)‘𝐶))) |
48 | 41, 3 | sseldd 3996 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Poly1‘𝑆))) |
49 | 26 | fveq1d 6909 | . . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑁) = (((eval1‘𝑆) ↾ 𝐵)‘𝑁)) |
50 | 3 | fvresd 6927 | . . . . . . 7 ⊢ (𝜑 → (((eval1‘𝑆) ↾ 𝐵)‘𝑁) = ((eval1‘𝑆)‘𝑁)) |
51 | 49, 50 | eqtr2d 2776 | . . . . . 6 ⊢ (𝜑 → ((eval1‘𝑆)‘𝑁) = (𝑄‘𝑁)) |
52 | 51 | fveq1d 6909 | . . . . 5 ⊢ (𝜑 → (((eval1‘𝑆)‘𝑁)‘𝐶) = ((𝑄‘𝑁)‘𝐶)) |
53 | 48, 52 | jca 511 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ (Base‘(Poly1‘𝑆)) ∧ (((eval1‘𝑆)‘𝑁)‘𝐶) = ((𝑄‘𝑁)‘𝐶))) |
54 | evls1muld.2 | . . . 4 ⊢ · = (.r‘𝑆) | |
55 | 24, 4, 23, 35, 25, 36, 47, 53, 15, 54 | evl1muld 22363 | . . 3 ⊢ (𝜑 → ((𝑀(.r‘(Poly1‘𝑆))𝑁) ∈ (Base‘(Poly1‘𝑆)) ∧ (((eval1‘𝑆)‘(𝑀(.r‘(Poly1‘𝑆))𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶)))) |
56 | 55 | simprd 495 | . 2 ⊢ (𝜑 → (((eval1‘𝑆)‘(𝑀(.r‘(Poly1‘𝑆))𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶))) |
57 | 21, 34, 56 | 3eqtr3d 2783 | 1 ⊢ (𝜑 → ((𝑄‘(𝑀 × 𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 .rcmulr 17299 Ringcrg 20251 CRingccrg 20252 SubRingcsubrg 20586 PwSer1cps1 22192 Poly1cpl1 22194 evalSub1 ces1 22333 eval1ce1 22334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-assa 21891 df-asp 21892 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-evls 22116 df-evl 22117 df-psr1 22197 df-vr1 22198 df-ply1 22199 df-coe1 22200 df-evls1 22335 df-evl1 22336 |
This theorem is referenced by: evls1maprhm 22396 irngnzply1lem 33705 minplyirred 33719 irredminply 33722 |
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