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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 24674 | . . 3 ⊢ 𝐷 = (1o mDeg 𝑅) |
3 | eqid 2821 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
4 | deg1leb.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | eqid 2821 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
6 | deg1leb.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
7 | 4, 5, 6 | ply1bas 20363 | . . 3 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅)) |
8 | deg1leb.y | . . 3 ⊢ 0 = (0g‘𝑅) | |
9 | psr1baslem 20353 | . . 3 ⊢ (ℕ0 ↑m 1o) = {𝑦 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑦 “ ℕ) ∈ Fin} | |
10 | tdeglem2 24655 | . . 3 ⊢ (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (ℂfld Σg 𝑥)) | |
11 | 2, 3, 7, 8, 9, 10 | mdegval 24657 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup(((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )), ℝ*, < )) |
12 | 8 | fvexi 6684 | . . . . . . . 8 ⊢ 0 ∈ V |
13 | suppimacnv 7841 | . . . . . . . 8 ⊢ ((𝐹 ∈ 𝐵 ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) | |
14 | 12, 13 | mpan2 689 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
15 | 14 | imaeq2d 5929 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) “ (◡𝐹 “ (V ∖ { 0 })))) |
16 | imaco 6104 | . . . . . 6 ⊢ (((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) “ (V ∖ { 0 })) = ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) “ (◡𝐹 “ (V ∖ { 0 }))) | |
17 | 15, 16 | syl6eqr 2874 | . . . . 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 8116 | . . . . . . . . . 10 ⊢ 1o = {∅} | |
20 | nn0ex 11904 | . . . . . . . . . 10 ⊢ ℕ0 ∈ V | |
21 | 0ex 5211 | . . . . . . . . . 10 ⊢ ∅ ∈ V | |
22 | eqid 2821 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) | |
23 | 19, 20, 21, 22 | mapsncnv 8457 | . . . . . . . . 9 ⊢ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦})) |
24 | 18, 6, 4, 23 | coe1fval2 20378 | . . . . . . . 8 ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)))) |
25 | 24 | cnveqd 5746 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → ◡𝐴 = ◡(𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)))) |
26 | cnvco 5756 | . . . . . . . 8 ⊢ ◡(𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅))) = (◡◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) | |
27 | cocnvcnv1 6110 | . . . . . . . 8 ⊢ (◡◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) = ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) | |
28 | 26, 27 | eqtri 2844 | . . . . . . 7 ⊢ ◡(𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅))) = ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) |
29 | 25, 28 | syl6req 2873 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) = ◡𝐴) |
30 | 29 | imaeq1d 5928 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → (((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) ∘ ◡𝐹) “ (V ∖ { 0 })) = (◡𝐴 “ (V ∖ { 0 }))) |
31 | 17, 30 | eqtrd 2856 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = (◡𝐴 “ (V ∖ { 0 }))) |
32 | 18 | fvexi 6684 | . . . . 5 ⊢ 𝐴 ∈ V |
33 | suppimacnv 7841 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 0 ∈ V) → (𝐴 supp 0 ) = (◡𝐴 “ (V ∖ { 0 }))) | |
34 | 33 | eqcomd 2827 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 0 ∈ V) → (◡𝐴 “ (V ∖ { 0 })) = (𝐴 supp 0 )) |
35 | 32, 12, 34 | mp2an 690 | . . . 4 ⊢ (◡𝐴 “ (V ∖ { 0 })) = (𝐴 supp 0 ) |
36 | 31, 35 | syl6eq 2872 | . . 3 ⊢ (𝐹 ∈ 𝐵 → ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )) = (𝐴 supp 0 )) |
37 | 36 | supeq1d 8910 | . 2 ⊢ (𝐹 ∈ 𝐵 → sup(((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) “ (𝐹 supp 0 )), ℝ*, < ) = sup((𝐴 supp 0 ), ℝ*, < )) |
38 | 11, 37 | eqtrd 2856 | 1 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐴 supp 0 ), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∖ cdif 3933 ∅c0 4291 {csn 4567 ↦ cmpt 5146 ◡ccnv 5554 “ cima 5558 ∘ ccom 5559 ‘cfv 6355 (class class class)co 7156 supp csupp 7830 1oc1o 8095 ↑m cmap 8406 supcsup 8904 ℝ*cxr 10674 < clt 10675 ℕ0cn0 11898 Basecbs 16483 0gc0g 16713 mPoly cmpl 20133 PwSer1cps1 20343 Poly1cpl1 20345 coe1cco1 20346 deg1 cdg1 24648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-0g 16715 df-gsum 16716 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-mgp 19240 df-ring 19299 df-cring 19300 df-psr 20136 df-mpl 20138 df-opsr 20140 df-psr1 20348 df-ply1 20350 df-coe1 20351 df-cnfld 20546 df-mdeg 24649 df-deg1 24650 |
This theorem is referenced by: deg1mul3 24709 deg1mul3le 24710 |
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