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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 8422 | . . . . . . 7 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ℎ:{∅}⟶ℕ0) | |
2 | 1 | feqmptd 6728 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ℎ = (𝑥 ∈ {∅} ↦ (ℎ‘𝑥))) |
3 | 2 | oveq2d 7166 | . . . . 5 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℂfld Σg ℎ) = (ℂfld Σg (𝑥 ∈ {∅} ↦ (ℎ‘𝑥)))) |
4 | cnring 20561 | . . . . . . 7 ⊢ ℂfld ∈ Ring | |
5 | ringmnd 19300 | . . . . . . 7 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
6 | 4, 5 | mp1i 13 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ℂfld ∈ Mnd) |
7 | 0ex 5204 | . . . . . . 7 ⊢ ∅ ∈ V | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ∅ ∈ V) |
9 | 7 | snid 4595 | . . . . . . . 8 ⊢ ∅ ∈ {∅} |
10 | ffvelrn 6844 | . . . . . . . 8 ⊢ ((ℎ:{∅}⟶ℕ0 ∧ ∅ ∈ {∅}) → (ℎ‘∅) ∈ ℕ0) | |
11 | 1, 9, 10 | sylancl 588 | . . . . . . 7 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℎ‘∅) ∈ ℕ0) |
12 | 11 | nn0cnd 11951 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℎ‘∅) ∈ ℂ) |
13 | cnfldbas 20543 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
14 | fveq2 6665 | . . . . . . 7 ⊢ (𝑥 = ∅ → (ℎ‘𝑥) = (ℎ‘∅)) | |
15 | 13, 14 | gsumsn 19068 | . . . . . 6 ⊢ ((ℂfld ∈ Mnd ∧ ∅ ∈ V ∧ (ℎ‘∅) ∈ ℂ) → (ℂfld Σg (𝑥 ∈ {∅} ↦ (ℎ‘𝑥))) = (ℎ‘∅)) |
16 | 6, 8, 12, 15 | syl3anc 1367 | . . . . 5 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℂfld Σg (𝑥 ∈ {∅} ↦ (ℎ‘𝑥))) = (ℎ‘∅)) |
17 | 3, 16 | eqtrd 2856 | . . . 4 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℂfld Σg ℎ) = (ℎ‘∅)) |
18 | df1o2 8110 | . . . . 5 ⊢ 1o = {∅} | |
19 | 18 | oveq2i 7161 | . . . 4 ⊢ (ℕ0 ↑m 1o) = (ℕ0 ↑m {∅}) |
20 | 17, 19 | eleq2s 2931 | . . 3 ⊢ (ℎ ∈ (ℕ0 ↑m 1o) → (ℂfld Σg ℎ) = (ℎ‘∅)) |
21 | 20 | eqcomd 2827 | . 2 ⊢ (ℎ ∈ (ℕ0 ↑m 1o) → (ℎ‘∅) = (ℂfld Σg ℎ)) |
22 | 21 | mpteq2ia 5150 | 1 ⊢ (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℎ‘∅)) = (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℂfld Σg ℎ)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∅c0 4291 {csn 4561 ↦ cmpt 5139 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 1oc1o 8089 ↑m cmap 8400 ℂcc 10529 ℕ0cn0 11891 Σg cgsu 16708 Mndcmnd 17905 Ringcrg 19291 ℂfldccnfld 20539 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-gsum 16710 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-mgp 19234 df-ring 19293 df-cring 19294 df-cnfld 20540 |
This theorem is referenced by: deg1ldg 24680 deg1leb 24683 deg1val 24684 |
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