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Mirrors > Home > MPE Home > Th. List > deg1mul3 | Structured version Visualization version GIF version |
Description: Degree of multiplication of a polynomial on the left by a nonzero-dividing scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof shortened by AV, 25-Jul-2019.) |
Ref | Expression |
---|---|
deg1mul3.d | ⊢ 𝐷 = (deg1‘𝑅) |
deg1mul3.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1mul3.e | ⊢ 𝐸 = (RLReg‘𝑅) |
deg1mul3.b | ⊢ 𝐵 = (Base‘𝑃) |
deg1mul3.t | ⊢ · = (.r‘𝑃) |
deg1mul3.a | ⊢ 𝐴 = (algSc‘𝑃) |
Ref | Expression |
---|---|
deg1mul3 | ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐷‘((𝐴‘𝐹) · 𝐺)) = (𝐷‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1mul3.e | . . . . . . . 8 ⊢ 𝐸 = (RLReg‘𝑅) | |
2 | eqid 2735 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | rrgss 20719 | . . . . . . 7 ⊢ 𝐸 ⊆ (Base‘𝑅) |
4 | 3 | sseli 3991 | . . . . . 6 ⊢ (𝐹 ∈ 𝐸 → 𝐹 ∈ (Base‘𝑅)) |
5 | deg1mul3.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
6 | deg1mul3.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑃) | |
7 | deg1mul3.a | . . . . . . 7 ⊢ 𝐴 = (algSc‘𝑃) | |
8 | deg1mul3.t | . . . . . . 7 ⊢ · = (.r‘𝑃) | |
9 | eqid 2735 | . . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
10 | 5, 6, 2, 7, 8, 9 | coe1sclmul 22301 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ (Base‘𝑅) ∧ 𝐺 ∈ 𝐵) → (coe1‘((𝐴‘𝐹) · 𝐺)) = ((ℕ0 × {𝐹}) ∘f (.r‘𝑅)(coe1‘𝐺))) |
11 | 4, 10 | syl3an2 1163 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (coe1‘((𝐴‘𝐹) · 𝐺)) = ((ℕ0 × {𝐹}) ∘f (.r‘𝑅)(coe1‘𝐺))) |
12 | 11 | oveq1d 7446 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → ((coe1‘((𝐴‘𝐹) · 𝐺)) supp (0g‘𝑅)) = (((ℕ0 × {𝐹}) ∘f (.r‘𝑅)(coe1‘𝐺)) supp (0g‘𝑅))) |
13 | eqid 2735 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
14 | nn0ex 12530 | . . . . . 6 ⊢ ℕ0 ∈ V | |
15 | 14 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → ℕ0 ∈ V) |
16 | simp1 1135 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝑅 ∈ Ring) | |
17 | simp2 1136 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐸) | |
18 | eqid 2735 | . . . . . . 7 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
19 | 18, 6, 5, 2 | coe1f 22229 | . . . . . 6 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
20 | 19 | 3ad2ant3 1134 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
21 | 1, 2, 9, 13, 15, 16, 17, 20 | rrgsupp 20718 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (((ℕ0 × {𝐹}) ∘f (.r‘𝑅)(coe1‘𝐺)) supp (0g‘𝑅)) = ((coe1‘𝐺) supp (0g‘𝑅))) |
22 | 12, 21 | eqtrd 2775 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → ((coe1‘((𝐴‘𝐹) · 𝐺)) supp (0g‘𝑅)) = ((coe1‘𝐺) supp (0g‘𝑅))) |
23 | 22 | supeq1d 9484 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → sup(((coe1‘((𝐴‘𝐹) · 𝐺)) supp (0g‘𝑅)), ℝ*, < ) = sup(((coe1‘𝐺) supp (0g‘𝑅)), ℝ*, < )) |
24 | 5 | ply1ring 22265 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
25 | 24 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝑃 ∈ Ring) |
26 | 5, 7, 2, 6 | ply1sclf 22304 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝐴:(Base‘𝑅)⟶𝐵) |
27 | 26 | 3ad2ant1 1132 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝐴:(Base‘𝑅)⟶𝐵) |
28 | 4 | 3ad2ant2 1133 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ (Base‘𝑅)) |
29 | 27, 28 | ffvelcdmd 7105 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐴‘𝐹) ∈ 𝐵) |
30 | simp3 1137 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵) | |
31 | 6, 8 | ringcl 20268 | . . . 4 ⊢ ((𝑃 ∈ Ring ∧ (𝐴‘𝐹) ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐴‘𝐹) · 𝐺) ∈ 𝐵) |
32 | 25, 29, 30, 31 | syl3anc 1370 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → ((𝐴‘𝐹) · 𝐺) ∈ 𝐵) |
33 | deg1mul3.d | . . . 4 ⊢ 𝐷 = (deg1‘𝑅) | |
34 | eqid 2735 | . . . 4 ⊢ (coe1‘((𝐴‘𝐹) · 𝐺)) = (coe1‘((𝐴‘𝐹) · 𝐺)) | |
35 | 33, 5, 6, 13, 34 | deg1val 26150 | . . 3 ⊢ (((𝐴‘𝐹) · 𝐺) ∈ 𝐵 → (𝐷‘((𝐴‘𝐹) · 𝐺)) = sup(((coe1‘((𝐴‘𝐹) · 𝐺)) supp (0g‘𝑅)), ℝ*, < )) |
36 | 32, 35 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐷‘((𝐴‘𝐹) · 𝐺)) = sup(((coe1‘((𝐴‘𝐹) · 𝐺)) supp (0g‘𝑅)), ℝ*, < )) |
37 | 33, 5, 6, 13, 18 | deg1val 26150 | . . 3 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) = sup(((coe1‘𝐺) supp (0g‘𝑅)), ℝ*, < )) |
38 | 37 | 3ad2ant3 1134 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐷‘𝐺) = sup(((coe1‘𝐺) supp (0g‘𝑅)), ℝ*, < )) |
39 | 23, 36, 38 | 3eqtr4d 2785 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐷‘((𝐴‘𝐹) · 𝐺)) = (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 × cxp 5687 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 supp csupp 8184 supcsup 9478 ℝ*cxr 11292 < clt 11293 ℕ0cn0 12524 Basecbs 17245 .rcmulr 17299 0gc0g 17486 Ringcrg 20251 RLRegcrlreg 20708 algSccascl 21890 Poly1cpl1 22194 coe1cco1 22195 deg1cdg1 26108 |
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-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-rlreg 20711 df-lmod 20877 df-lss 20948 df-cnfld 21383 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-psr1 22197 df-vr1 22198 df-ply1 22199 df-coe1 22200 df-mdeg 26109 df-deg1 26110 |
This theorem is referenced by: uc1pmon1p 26206 ig1peu 26229 |
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