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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 8868 | . . . . . . 7 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ℎ:{∅}⟶ℕ0) | |
2 | 1 | feqmptd 6967 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ℎ = (𝑥 ∈ {∅} ↦ (ℎ‘𝑥))) |
3 | 2 | oveq2d 7436 | . . . . 5 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℂfld Σg ℎ) = (ℂfld Σg (𝑥 ∈ {∅} ↦ (ℎ‘𝑥)))) |
4 | cnring 21318 | . . . . . . 7 ⊢ ℂfld ∈ Ring | |
5 | ringmnd 20183 | . . . . . . 7 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
6 | 4, 5 | mp1i 13 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ℂfld ∈ Mnd) |
7 | 0ex 5307 | . . . . . . 7 ⊢ ∅ ∈ V | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → ∅ ∈ V) |
9 | 7 | snid 4665 | . . . . . . . 8 ⊢ ∅ ∈ {∅} |
10 | ffvelcdm 7091 | . . . . . . . 8 ⊢ ((ℎ:{∅}⟶ℕ0 ∧ ∅ ∈ {∅}) → (ℎ‘∅) ∈ ℕ0) | |
11 | 1, 9, 10 | sylancl 585 | . . . . . . 7 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℎ‘∅) ∈ ℕ0) |
12 | 11 | nn0cnd 12565 | . . . . . 6 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℎ‘∅) ∈ ℂ) |
13 | cnfldbas 21283 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
14 | fveq2 6897 | . . . . . . 7 ⊢ (𝑥 = ∅ → (ℎ‘𝑥) = (ℎ‘∅)) | |
15 | 13, 14 | gsumsn 19909 | . . . . . 6 ⊢ ((ℂfld ∈ Mnd ∧ ∅ ∈ V ∧ (ℎ‘∅) ∈ ℂ) → (ℂfld Σg (𝑥 ∈ {∅} ↦ (ℎ‘𝑥))) = (ℎ‘∅)) |
16 | 6, 8, 12, 15 | syl3anc 1369 | . . . . 5 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℂfld Σg (𝑥 ∈ {∅} ↦ (ℎ‘𝑥))) = (ℎ‘∅)) |
17 | 3, 16 | eqtrd 2768 | . . . 4 ⊢ (ℎ ∈ (ℕ0 ↑m {∅}) → (ℂfld Σg ℎ) = (ℎ‘∅)) |
18 | df1o2 8494 | . . . . 5 ⊢ 1o = {∅} | |
19 | 18 | oveq2i 7431 | . . . 4 ⊢ (ℕ0 ↑m 1o) = (ℕ0 ↑m {∅}) |
20 | 17, 19 | eleq2s 2847 | . . 3 ⊢ (ℎ ∈ (ℕ0 ↑m 1o) → (ℂfld Σg ℎ) = (ℎ‘∅)) |
21 | 20 | eqcomd 2734 | . 2 ⊢ (ℎ ∈ (ℕ0 ↑m 1o) → (ℎ‘∅) = (ℂfld Σg ℎ)) |
22 | 21 | mpteq2ia 5251 | 1 ⊢ (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℎ‘∅)) = (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℂfld Σg ℎ)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∅c0 4323 {csn 4629 ↦ cmpt 5231 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 1oc1o 8480 ↑m cmap 8845 ℂcc 11137 ℕ0cn0 12503 Σg cgsu 17422 Mndcmnd 18694 Ringcrg 20173 ℂfldccnfld 21279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-addf 11218 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-mulr 17247 df-starv 17248 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-0g 17423 df-gsum 17424 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-mulg 19024 df-cntz 19268 df-cmn 19737 df-mgp 20075 df-ring 20175 df-cring 20176 df-cnfld 21280 |
This theorem is referenced by: deg1ldg 26041 deg1leb 26044 deg1val 26045 |
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