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Mirrors > Home > MPE Home > Th. List > Mathboxes > mhphf2 | Structured version Visualization version GIF version |
Description: A homogeneous polynomial
defines a homogeneous function; this is mhphf 42584
with simpler notation in the conclusion in exchange for a complex
definition of ∙, which is
based on frlmvscafval 21804 but without the
finite support restriction (frlmpws 21788, frlmbas 21793) on the assignments
𝐴 from variables to values.
TODO?: Polynomials (df-mpl 21949) are defined to have a finite amount of terms (of finite degree). As such, any assignment may be replaced by an assignment with finite support (as only a finite amount of variables matter in a given polynomial, even if the set of variables is infinite). So the finite support restriction can be assumed without loss of generality. (Contributed by SN, 11-Nov-2024.) |
Ref | Expression |
---|---|
mhphf2.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) |
mhphf2.h | ⊢ 𝐻 = (𝐼 mHomP 𝑈) |
mhphf2.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
mhphf2.k | ⊢ 𝐾 = (Base‘𝑆) |
mhphf2.b | ⊢ ∙ = ( ·𝑠 ‘((ringLMod‘𝑆) ↑s 𝐼)) |
mhphf2.m | ⊢ · = (.r‘𝑆) |
mhphf2.e | ⊢ ↑ = (.g‘(mulGrp‘𝑆)) |
mhphf2.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
mhphf2.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
mhphf2.l | ⊢ (𝜑 → 𝐿 ∈ 𝑅) |
mhphf2.x | ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) |
mhphf2.a | ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) |
Ref | Expression |
---|---|
mhphf2 | ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . . . 5 ⊢ ((ringLMod‘𝑆) ↑s 𝐼) = ((ringLMod‘𝑆) ↑s 𝐼) | |
2 | eqid 2735 | . . . . 5 ⊢ (Base‘((ringLMod‘𝑆) ↑s 𝐼)) = (Base‘((ringLMod‘𝑆) ↑s 𝐼)) | |
3 | rlmvsca 21225 | . . . . 5 ⊢ (.r‘𝑆) = ( ·𝑠 ‘(ringLMod‘𝑆)) | |
4 | mhphf2.b | . . . . 5 ⊢ ∙ = ( ·𝑠 ‘((ringLMod‘𝑆) ↑s 𝐼)) | |
5 | eqid 2735 | . . . . 5 ⊢ (Scalar‘(ringLMod‘𝑆)) = (Scalar‘(ringLMod‘𝑆)) | |
6 | eqid 2735 | . . . . 5 ⊢ (Base‘(Scalar‘(ringLMod‘𝑆))) = (Base‘(Scalar‘(ringLMod‘𝑆))) | |
7 | fvexd 6922 | . . . . 5 ⊢ (𝜑 → (ringLMod‘𝑆) ∈ V) | |
8 | reldmmhp 22159 | . . . . . 6 ⊢ Rel dom mHomP | |
9 | mhphf2.h | . . . . . 6 ⊢ 𝐻 = (𝐼 mHomP 𝑈) | |
10 | mhphf2.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) | |
11 | 8, 9, 10 | elfvov1 7473 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ V) |
12 | mhphf2.r | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
13 | mhphf2.k | . . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑆) | |
14 | 13 | subrgss 20589 | . . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐾) |
15 | 12, 14 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ⊆ 𝐾) |
16 | mhphf2.l | . . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ 𝑅) | |
17 | 15, 16 | sseldd 3996 | . . . . . 6 ⊢ (𝜑 → 𝐿 ∈ 𝐾) |
18 | mhphf2.s | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
19 | rlmsca 21223 | . . . . . . . . 9 ⊢ (𝑆 ∈ CRing → 𝑆 = (Scalar‘(ringLMod‘𝑆))) | |
20 | 18, 19 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Scalar‘(ringLMod‘𝑆))) |
21 | 20 | fveq2d 6911 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝑆) = (Base‘(Scalar‘(ringLMod‘𝑆)))) |
22 | 13, 21 | eqtrid 2787 | . . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘(ringLMod‘𝑆)))) |
23 | 17, 22 | eleqtrd 2841 | . . . . 5 ⊢ (𝜑 → 𝐿 ∈ (Base‘(Scalar‘(ringLMod‘𝑆)))) |
24 | mhphf2.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) | |
25 | 13 | oveq1i 7441 | . . . . . . 7 ⊢ (𝐾 ↑m 𝐼) = ((Base‘𝑆) ↑m 𝐼) |
26 | 24, 25 | eleqtrdi 2849 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ((Base‘𝑆) ↑m 𝐼)) |
27 | rlmbas 21218 | . . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘(ringLMod‘𝑆)) | |
28 | 1, 27 | pwsbas 17534 | . . . . . . 7 ⊢ (((ringLMod‘𝑆) ∈ V ∧ 𝐼 ∈ V) → ((Base‘𝑆) ↑m 𝐼) = (Base‘((ringLMod‘𝑆) ↑s 𝐼))) |
29 | 7, 11, 28 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → ((Base‘𝑆) ↑m 𝐼) = (Base‘((ringLMod‘𝑆) ↑s 𝐼))) |
30 | 26, 29 | eleqtrd 2841 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘((ringLMod‘𝑆) ↑s 𝐼))) |
31 | 1, 2, 3, 4, 5, 6, 7, 11, 23, 30 | pwsvscafval 17541 | . . . 4 ⊢ (𝜑 → (𝐿 ∙ 𝐴) = ((𝐼 × {𝐿}) ∘f (.r‘𝑆)𝐴)) |
32 | mhphf2.m | . . . . . . 7 ⊢ · = (.r‘𝑆) | |
33 | 32 | eqcomi 2744 | . . . . . 6 ⊢ (.r‘𝑆) = · |
34 | ofeq 7700 | . . . . . 6 ⊢ ((.r‘𝑆) = · → ∘f (.r‘𝑆) = ∘f · ) | |
35 | 33, 34 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ∘f (.r‘𝑆) = ∘f · ) |
36 | 35 | oveqd 7448 | . . . 4 ⊢ (𝜑 → ((𝐼 × {𝐿}) ∘f (.r‘𝑆)𝐴) = ((𝐼 × {𝐿}) ∘f · 𝐴)) |
37 | 31, 36 | eqtrd 2775 | . . 3 ⊢ (𝜑 → (𝐿 ∙ 𝐴) = ((𝐼 × {𝐿}) ∘f · 𝐴)) |
38 | 37 | fveq2d 6911 | . 2 ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘f · 𝐴))) |
39 | mhphf2.q | . . 3 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
40 | mhphf2.u | . . 3 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
41 | mhphf2.e | . . 3 ⊢ ↑ = (.g‘(mulGrp‘𝑆)) | |
42 | 39, 9, 40, 13, 32, 41, 18, 12, 16, 10, 24 | mhphf 42584 | . 2 ⊢ (𝜑 → ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴))) |
43 | 38, 42 | eqtrd 2775 | 1 ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 {csn 4631 × cxp 5687 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 ↑m cmap 8865 Basecbs 17245 ↾s cress 17274 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 ↑s cpws 17493 .gcmg 19098 mulGrpcmgp 20152 CRingccrg 20252 SubRingcsubrg 20586 ringLModcrglmod 21189 evalSub ces 22114 mHomP cmhp 22151 |
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 ax-addf 11232 |
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-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 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-sra 21190 df-rgmod 21191 df-cnfld 21383 df-assa 21891 df-asp 21892 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-evls 22116 df-mhp 22158 |
This theorem is referenced by: (None) |
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