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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 8840 | . . . . . . 7 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ℎ:{∅}⟶ℕ0) | |
2 | 1 | feqmptd 6951 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ℎ = (𝑥 ∈ {∅} ↦ (ℎ‘𝑥))) |
3 | 2 | oveq2d 7418 | . . . . 5 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℂfld Σg ℎ) = (ℂfld Σg (𝑥 ∈ {∅} ↦ (ℎ‘𝑥)))) |
4 | cnring 21272 | . . . . . . 7 ⊢ ℂfld ∈ Ring | |
5 | ringmnd 20144 | . . . . . . 7 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
6 | 4, 5 | mp1i 13 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ℂfld ∈ Mnd) |
7 | 0ex 5298 | . . . . . . 7 ⊢ ∅ ∈ V | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ∅ ∈ V) |
9 | 7 | snid 4657 | . . . . . . . 8 ⊢ ∅ ∈ {∅} |
10 | ffvelcdm 7074 | . . . . . . . 8 ⊢ ((ℎ:{∅}⟶ℕ0 ∧ ∅ ∈ {∅}) → (ℎ‘∅) ∈ ℕ0) | |
11 | 1, 9, 10 | sylancl 585 | . . . . . . 7 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℎ‘∅) ∈ ℕ0) |
12 | 11 | nn0cnd 12533 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℎ‘∅) ∈ ℂ) |
13 | cnfldbas 21238 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
14 | fveq2 6882 | . . . . . . 7 ⊢ (𝑥 = ∅ → (ℎ‘𝑥) = (ℎ‘∅)) | |
15 | 13, 14 | gsumsn 19870 | . . . . . 6 ⊢ ((ℂfld ∈ Mnd ∧ ∅ ∈ V ∧ (ℎ‘∅) ∈ ℂ) → (ℂfld Σg (𝑥 ∈ {∅} ↦ (ℎ‘𝑥))) = (ℎ‘∅)) |
16 | 6, 8, 12, 15 | syl3anc 1368 | . . . . 5 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℂfld Σg (𝑥 ∈ {∅} ↦ (ℎ‘𝑥))) = (ℎ‘∅)) |
17 | 3, 16 | eqtrd 2764 | . . . 4 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℂfld Σg ℎ) = (ℎ‘∅)) |
18 | df1o2 8469 | . . . . 5 ⊢ 1o = {∅} | |
19 | 18 | oveq2i 7413 | . . . 4 ⊢ (ℕ0 ↑m 1o) = (ℕ0 ↑m {∅}) |
20 | 17, 19 | eleq2s 2843 | . . 3 ⊢ (ℎ ∈ (ℕ0 ↑m 1o) → (ℂfld Σg ℎ) = (ℎ‘∅)) |
21 | 20 | eqcomd 2730 | . 2 ⊢ (ℎ ∈ (ℕ0 ↑m 1o) → (ℎ‘∅) = (ℂfld Σg ℎ)) |
22 | 21 | mpteq2ia 5242 | 1 ⊢ (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℎ‘∅)) = (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℂfld Σg ℎ)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3466 ∅c0 4315 {csn 4621 ↦ cmpt 5222 ⟶wf 6530 ‘cfv 6534 (class class class)co 7402 1oc1o 8455 ↑m cmap 8817 ℂcc 11105 ℕ0cn0 12471 Σg cgsu 17391 Mndcmnd 18663 Ringcrg 20134 ℂfldccnfld 21234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-addf 11186 ax-mulf 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-fzo 13629 df-seq 13968 df-hash 14292 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-mulr 17216 df-starv 17217 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-0g 17392 df-gsum 17393 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-mulg 18992 df-cntz 19229 df-cmn 19698 df-mgp 20036 df-ring 20136 df-cring 20137 df-cnfld 21235 |
This theorem is referenced by: deg1ldg 25972 deg1leb 25975 deg1val 25976 |
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