Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > tdeglem2 | Structured version Visualization version GIF version |
Description: Simplification of total degree for the univariate case. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
Ref | Expression |
---|---|
tdeglem2 | ⊢ (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℎ‘∅)) = (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℂfld Σg ℎ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 8460 | . . . . . . 7 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ℎ:{∅}⟶ℕ0) | |
2 | 1 | feqmptd 6738 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ℎ = (𝑥 ∈ {∅} ↦ (ℎ‘𝑥))) |
3 | 2 | oveq2d 7187 | . . . . 5 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℂfld Σg ℎ) = (ℂfld Σg (𝑥 ∈ {∅} ↦ (ℎ‘𝑥)))) |
4 | cnring 20240 | . . . . . . 7 ⊢ ℂfld ∈ Ring | |
5 | ringmnd 19427 | . . . . . . 7 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
6 | 4, 5 | mp1i 13 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ℂfld ∈ Mnd) |
7 | 0ex 5176 | . . . . . . 7 ⊢ ∅ ∈ V | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ∅ ∈ V) |
9 | 7 | snid 4553 | . . . . . . . 8 ⊢ ∅ ∈ {∅} |
10 | ffvelrn 6860 | . . . . . . . 8 ⊢ ((ℎ:{∅}⟶ℕ0 ∧ ∅ ∈ {∅}) → (ℎ‘∅) ∈ ℕ0) | |
11 | 1, 9, 10 | sylancl 589 | . . . . . . 7 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℎ‘∅) ∈ ℕ0) |
12 | 11 | nn0cnd 12039 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℎ‘∅) ∈ ℂ) |
13 | cnfldbas 20222 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
14 | fveq2 6675 | . . . . . . 7 ⊢ (𝑥 = ∅ → (ℎ‘𝑥) = (ℎ‘∅)) | |
15 | 13, 14 | gsumsn 19194 | . . . . . 6 ⊢ ((ℂfld ∈ Mnd ∧ ∅ ∈ V ∧ (ℎ‘∅) ∈ ℂ) → (ℂfld Σg (𝑥 ∈ {∅} ↦ (ℎ‘𝑥))) = (ℎ‘∅)) |
16 | 6, 8, 12, 15 | syl3anc 1372 | . . . . 5 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℂfld Σg (𝑥 ∈ {∅} ↦ (ℎ‘𝑥))) = (ℎ‘∅)) |
17 | 3, 16 | eqtrd 2773 | . . . 4 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℂfld Σg ℎ) = (ℎ‘∅)) |
18 | df1o2 8144 | . . . . 5 ⊢ 1o = {∅} | |
19 | 18 | oveq2i 7182 | . . . 4 ⊢ (ℕ0 ↑m 1o) = (ℕ0 ↑m {∅}) |
20 | 17, 19 | eleq2s 2851 | . . 3 ⊢ (ℎ ∈ (ℕ0 ↑m 1o) → (ℂfld Σg ℎ) = (ℎ‘∅)) |
21 | 20 | eqcomd 2744 | . 2 ⊢ (ℎ ∈ (ℕ0 ↑m 1o) → (ℎ‘∅) = (ℂfld Σg ℎ)) |
22 | 21 | mpteq2ia 5122 | 1 ⊢ (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℎ‘∅)) = (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℂfld Σg ℎ)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2113 Vcvv 3398 ∅c0 4212 {csn 4517 ↦ cmpt 5111 ⟶wf 6336 ‘cfv 6340 (class class class)co 7171 1oc1o 8125 ↑m cmap 8438 ℂcc 10614 ℕ0cn0 11977 Σg cgsu 16818 Mndcmnd 18028 Ringcrg 19417 ℂfldccnfld 20218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-resscn 10673 ax-1cn 10674 ax-icn 10675 ax-addcl 10676 ax-addrcl 10677 ax-mulcl 10678 ax-mulrcl 10679 ax-mulcom 10680 ax-addass 10681 ax-mulass 10682 ax-distr 10683 ax-i2m1 10684 ax-1ne0 10685 ax-1rid 10686 ax-rnegex 10687 ax-rrecex 10688 ax-cnre 10689 ax-pre-lttri 10690 ax-pre-lttrn 10691 ax-pre-ltadd 10692 ax-pre-mulgt0 10693 ax-addf 10695 ax-mulf 10696 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7128 df-ov 7174 df-oprab 7175 df-mpo 7176 df-om 7601 df-1st 7715 df-2nd 7716 df-supp 7858 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-1o 8132 df-er 8321 df-map 8440 df-en 8557 df-dom 8558 df-sdom 8559 df-fin 8560 df-oi 9048 df-card 9442 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 df-sub 10951 df-neg 10952 df-nn 11718 df-2 11780 df-3 11781 df-4 11782 df-5 11783 df-6 11784 df-7 11785 df-8 11786 df-9 11787 df-n0 11978 df-z 12064 df-dec 12181 df-uz 12326 df-fz 12983 df-fzo 13126 df-seq 13462 df-hash 13784 df-struct 16589 df-ndx 16590 df-slot 16591 df-base 16593 df-sets 16594 df-plusg 16682 df-mulr 16683 df-starv 16684 df-tset 16688 df-ple 16689 df-ds 16691 df-unif 16692 df-0g 16819 df-gsum 16820 df-mgm 17969 df-sgrp 18018 df-mnd 18029 df-grp 18223 df-mulg 18344 df-cntz 18566 df-cmn 19027 df-mgp 19360 df-ring 19419 df-cring 19420 df-cnfld 20219 |
This theorem is referenced by: deg1ldg 24845 deg1leb 24848 deg1val 24849 |
Copyright terms: Public domain | W3C validator |