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Mirrors > Home > MPE Home > Th. List > deg1val | Structured version Visualization version GIF version |
Description: Value of the univariate degree as a supremum. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Jul-2019.) |
Ref | Expression |
---|---|
deg1leb.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1leb.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1leb.b | ⊢ 𝐵 = (Base‘𝑃) |
deg1leb.y | ⊢ 0 = (0g‘𝑅) |
deg1leb.a | ⊢ 𝐴 = (coe1‘𝐹) |
Ref | Expression |
---|---|
deg1val | ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐴 supp 0 ), ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1leb.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
2 | 1 | deg1fval 24239 | . . 3 ⊢ 𝐷 = (1o mDeg 𝑅) |
3 | eqid 2825 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
4 | deg1leb.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | eqid 2825 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
6 | deg1leb.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
7 | 4, 5, 6 | ply1bas 19925 | . . 3 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅)) |
8 | deg1leb.y | . . 3 ⊢ 0 = (0g‘𝑅) | |
9 | psr1baslem 19915 | . . 3 ⊢ (ℕ0 ↑𝑚 1o) = {𝑦 ∈ (ℕ0 ↑𝑚 1o) ∣ (◡𝑦 “ ℕ) ∈ Fin} | |
10 | tdeglem2 24220 | . . 3 ⊢ (𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (ℂfld Σg 𝑥)) | |
11 | 2, 3, 7, 8, 9, 10 | mdegval 24222 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup(((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )), ℝ*, < )) |
12 | 8 | fvexi 6447 | . . . . . . . 8 ⊢ 0 ∈ V |
13 | suppimacnv 7570 | . . . . . . . 8 ⊢ ((𝐹 ∈ 𝐵 ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) | |
14 | 12, 13 | mpan2 684 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
15 | 14 | imaeq2d 5707 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) “ (◡𝐹 “ (V ∖ { 0 })))) |
16 | imaco 5881 | . . . . . 6 ⊢ (((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) “ (V ∖ { 0 })) = ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) “ (◡𝐹 “ (V ∖ { 0 }))) | |
17 | 15, 16 | syl6eqr 2879 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = (((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) “ (V ∖ { 0 }))) |
18 | deg1leb.a | . . . . . . . . 9 ⊢ 𝐴 = (coe1‘𝐹) | |
19 | df1o2 7839 | . . . . . . . . . 10 ⊢ 1o = {∅} | |
20 | nn0ex 11625 | . . . . . . . . . 10 ⊢ ℕ0 ∈ V | |
21 | 0ex 5014 | . . . . . . . . . 10 ⊢ ∅ ∈ V | |
22 | eqid 2825 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) | |
23 | 19, 20, 21, 22 | mapsncnv 8171 | . . . . . . . . 9 ⊢ ◡(𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦})) |
24 | 18, 6, 4, 23 | coe1fval2 19940 | . . . . . . . 8 ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)))) |
25 | 24 | cnveqd 5530 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → ◡𝐴 = ◡(𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)))) |
26 | cnvco 5540 | . . . . . . . 8 ⊢ ◡(𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅))) = (◡◡(𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) | |
27 | cocnvcnv1 5887 | . . . . . . . 8 ⊢ (◡◡(𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) = ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) | |
28 | 26, 27 | eqtri 2849 | . . . . . . 7 ⊢ ◡(𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅))) = ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) |
29 | 25, 28 | syl6req 2878 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) = ◡𝐴) |
30 | 29 | imaeq1d 5706 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → (((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) “ (V ∖ { 0 })) = (◡𝐴 “ (V ∖ { 0 }))) |
31 | 17, 30 | eqtrd 2861 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = (◡𝐴 “ (V ∖ { 0 }))) |
32 | 18 | fvexi 6447 | . . . . 5 ⊢ 𝐴 ∈ V |
33 | suppimacnv 7570 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 0 ∈ V) → (𝐴 supp 0 ) = (◡𝐴 “ (V ∖ { 0 }))) | |
34 | 33 | eqcomd 2831 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 0 ∈ V) → (◡𝐴 “ (V ∖ { 0 })) = (𝐴 supp 0 )) |
35 | 32, 12, 34 | mp2an 685 | . . . 4 ⊢ (◡𝐴 “ (V ∖ { 0 })) = (𝐴 supp 0 ) |
36 | 31, 35 | syl6eq 2877 | . . 3 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = (𝐴 supp 0 )) |
37 | 36 | supeq1d 8621 | . 2 ⊢ (𝐹 ∈ 𝐵 → sup(((𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )), ℝ*, < ) = sup((𝐴 supp 0 ), ℝ*, < )) |
38 | 11, 37 | eqtrd 2861 | 1 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐴 supp 0 ), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 Vcvv 3414 ∖ cdif 3795 ∅c0 4144 {csn 4397 ↦ cmpt 4952 ◡ccnv 5341 “ cima 5345 ∘ ccom 5346 ‘cfv 6123 (class class class)co 6905 supp csupp 7559 1oc1o 7819 ↑𝑚 cmap 8122 supcsup 8615 ℝ*cxr 10390 < clt 10391 ℕ0cn0 11618 Basecbs 16222 0gc0g 16453 mPoly cmpl 19714 PwSer1cps1 19905 Poly1cpl1 19907 coe1cco1 19908 deg1 cdg1 24213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-addf 10331 ax-mulf 10332 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-sup 8617 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-fz 12620 df-fzo 12761 df-seq 13096 df-hash 13411 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-starv 16320 df-sca 16321 df-vsca 16322 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-0g 16455 df-gsum 16456 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-mulg 17895 df-cntz 18100 df-cmn 18548 df-mgp 18844 df-ring 18903 df-cring 18904 df-psr 19717 df-mpl 19719 df-opsr 19721 df-psr1 19910 df-ply1 19912 df-coe1 19913 df-cnfld 20107 df-mdeg 24214 df-deg1 24215 |
This theorem is referenced by: deg1mul3 24274 deg1mul3le 24275 |
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