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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 8417 | . . . . . . 7 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ℎ:{∅}⟶ℕ0) | |
2 | 1 | feqmptd 6726 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ℎ = (𝑥 ∈ {∅} ↦ (ℎ‘𝑥))) |
3 | 2 | oveq2d 7161 | . . . . 5 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℂfld Σg ℎ) = (ℂfld Σg (𝑥 ∈ {∅} ↦ (ℎ‘𝑥)))) |
4 | cnring 20495 | . . . . . . 7 ⊢ ℂfld ∈ Ring | |
5 | ringmnd 19235 | . . . . . . 7 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
6 | 4, 5 | mp1i 13 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ℂfld ∈ Mnd) |
7 | 0ex 5202 | . . . . . . 7 ⊢ ∅ ∈ V | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ∅ ∈ V) |
9 | 7 | snid 4591 | . . . . . . . 8 ⊢ ∅ ∈ {∅} |
10 | ffvelrn 6841 | . . . . . . . 8 ⊢ ((ℎ:{∅}⟶ℕ0 ∧ ∅ ∈ {∅}) → (ℎ‘∅) ∈ ℕ0) | |
11 | 1, 9, 10 | sylancl 586 | . . . . . . 7 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℎ‘∅) ∈ ℕ0) |
12 | 11 | nn0cnd 11945 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℎ‘∅) ∈ ℂ) |
13 | cnfldbas 20477 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
14 | fveq2 6663 | . . . . . . 7 ⊢ (𝑥 = ∅ → (ℎ‘𝑥) = (ℎ‘∅)) | |
15 | 13, 14 | gsumsn 19003 | . . . . . 6 ⊢ ((ℂfld ∈ Mnd ∧ ∅ ∈ V ∧ (ℎ‘∅) ∈ ℂ) → (ℂfld Σg (𝑥 ∈ {∅} ↦ (ℎ‘𝑥))) = (ℎ‘∅)) |
16 | 6, 8, 12, 15 | syl3anc 1363 | . . . . 5 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℂfld Σg (𝑥 ∈ {∅} ↦ (ℎ‘𝑥))) = (ℎ‘∅)) |
17 | 3, 16 | eqtrd 2853 | . . . 4 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℂfld Σg ℎ) = (ℎ‘∅)) |
18 | df1o2 8105 | . . . . 5 ⊢ 1o = {∅} | |
19 | 18 | oveq2i 7156 | . . . 4 ⊢ (ℕ0 ↑m 1o) = (ℕ0 ↑m {∅}) |
20 | 17, 19 | eleq2s 2928 | . . 3 ⊢ (ℎ ∈ (ℕ0 ↑m 1o) → (ℂfld Σg ℎ) = (ℎ‘∅)) |
21 | 20 | eqcomd 2824 | . 2 ⊢ (ℎ ∈ (ℕ0 ↑m 1o) → (ℎ‘∅) = (ℂfld Σg ℎ)) |
22 | 21 | mpteq2ia 5148 | 1 ⊢ (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℎ‘∅)) = (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℂfld Σg ℎ)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 {csn 4557 ↦ cmpt 5137 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 1oc1o 8084 ↑m cmap 8395 ℂcc 10523 ℕ0cn0 11885 Σg cgsu 16702 Mndcmnd 17899 Ringcrg 19226 ℂfldccnfld 20473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-gsum 16704 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-mgp 19169 df-ring 19228 df-cring 19229 df-cnfld 20474 |
This theorem is referenced by: deg1ldg 24613 deg1leb 24616 deg1val 24617 |
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