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Mirrors > Home > MPE Home > Th. List > tdeglem3 | Structured version Visualization version GIF version |
Description: Additivity of the total degree helper function. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.) |
Ref | Expression |
---|---|
tdeglem.a | ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} |
tdeglem.h | ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) |
Ref | Expression |
---|---|
tdeglem3 | ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐻‘(𝑋 ∘f + 𝑌)) = ((𝐻‘𝑋) + (𝐻‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldbas 20392 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
2 | cnfld0 20412 | . . 3 ⊢ 0 = (0g‘ℂfld) | |
3 | cnfldadd 20393 | . . 3 ⊢ + = (+g‘ℂfld) | |
4 | cnring 20410 | . . . 4 ⊢ ℂfld ∈ Ring | |
5 | ringcmn 19624 | . . . 4 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd) | |
6 | 4, 5 | mp1i 13 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ℂfld ∈ CMnd) |
7 | simpl 486 | . . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ 𝐴) | |
8 | tdeglem.a | . . . . . . . 8 ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} | |
9 | 8 | psrbagf 20901 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐴 → 𝑋:𝐼⟶ℕ0) |
10 | nn0sscn 12120 | . . . . . . 7 ⊢ ℕ0 ⊆ ℂ | |
11 | fss 6581 | . . . . . . 7 ⊢ ((𝑋:𝐼⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑋:𝐼⟶ℂ) | |
12 | 9, 10, 11 | sylancl 589 | . . . . . 6 ⊢ (𝑋 ∈ 𝐴 → 𝑋:𝐼⟶ℂ) |
13 | 12 | adantr 484 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋:𝐼⟶ℂ) |
14 | 13 | ffnd 6565 | . . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 Fn 𝐼) |
15 | 7, 14 | fndmexd 7703 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝐼 ∈ V) |
16 | 8 | psrbagf 20901 | . . . . 5 ⊢ (𝑌 ∈ 𝐴 → 𝑌:𝐼⟶ℕ0) |
17 | fss 6581 | . . . . 5 ⊢ ((𝑌:𝐼⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑌:𝐼⟶ℂ) | |
18 | 16, 10, 17 | sylancl 589 | . . . 4 ⊢ (𝑌 ∈ 𝐴 → 𝑌:𝐼⟶ℂ) |
19 | 18 | adantl 485 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌:𝐼⟶ℂ) |
20 | 8 | psrbagfsupp 20903 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → 𝑋 finSupp 0) |
21 | 20 | adantr 484 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 finSupp 0) |
22 | 8 | psrbagfsupp 20903 | . . . 4 ⊢ (𝑌 ∈ 𝐴 → 𝑌 finSupp 0) |
23 | 22 | adantl 485 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌 finSupp 0) |
24 | 1, 2, 3, 6, 15, 13, 19, 21, 23 | gsumadd 19333 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (ℂfld Σg (𝑋 ∘f + 𝑌)) = ((ℂfld Σg 𝑋) + (ℂfld Σg 𝑌))) |
25 | 8 | psrbagaddcl 20911 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∘f + 𝑌) ∈ 𝐴) |
26 | oveq2 7240 | . . . 4 ⊢ (ℎ = (𝑋 ∘f + 𝑌) → (ℂfld Σg ℎ) = (ℂfld Σg (𝑋 ∘f + 𝑌))) | |
27 | tdeglem.h | . . . 4 ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) | |
28 | ovex 7265 | . . . 4 ⊢ (ℂfld Σg (𝑋 ∘f + 𝑌)) ∈ V | |
29 | 26, 27, 28 | fvmpt 6837 | . . 3 ⊢ ((𝑋 ∘f + 𝑌) ∈ 𝐴 → (𝐻‘(𝑋 ∘f + 𝑌)) = (ℂfld Σg (𝑋 ∘f + 𝑌))) |
30 | 25, 29 | syl 17 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐻‘(𝑋 ∘f + 𝑌)) = (ℂfld Σg (𝑋 ∘f + 𝑌))) |
31 | oveq2 7240 | . . . 4 ⊢ (ℎ = 𝑋 → (ℂfld Σg ℎ) = (ℂfld Σg 𝑋)) | |
32 | ovex 7265 | . . . 4 ⊢ (ℂfld Σg 𝑋) ∈ V | |
33 | 31, 27, 32 | fvmpt 6837 | . . 3 ⊢ (𝑋 ∈ 𝐴 → (𝐻‘𝑋) = (ℂfld Σg 𝑋)) |
34 | oveq2 7240 | . . . 4 ⊢ (ℎ = 𝑌 → (ℂfld Σg ℎ) = (ℂfld Σg 𝑌)) | |
35 | ovex 7265 | . . . 4 ⊢ (ℂfld Σg 𝑌) ∈ V | |
36 | 34, 27, 35 | fvmpt 6837 | . . 3 ⊢ (𝑌 ∈ 𝐴 → (𝐻‘𝑌) = (ℂfld Σg 𝑌)) |
37 | 33, 36 | oveqan12d 7251 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐻‘𝑋) + (𝐻‘𝑌)) = ((ℂfld Σg 𝑋) + (ℂfld Σg 𝑌))) |
38 | 24, 30, 37 | 3eqtr4d 2788 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐻‘(𝑋 ∘f + 𝑌)) = ((𝐻‘𝑋) + (𝐻‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 {crab 3066 Vcvv 3421 ⊆ wss 3881 class class class wbr 5068 ↦ cmpt 5150 ◡ccnv 5565 “ cima 5569 ⟶wf 6394 ‘cfv 6398 (class class class)co 7232 ∘f cof 7486 ↑m cmap 8529 Fincfn 8647 finSupp cfsupp 9010 ℂcc 10752 0cc0 10754 + caddc 10757 ℕcn 11855 ℕ0cn0 12115 Σg cgsu 16970 CMndccmn 19195 Ringcrg 19587 ℂfldccnfld 20388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5194 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 ax-cnex 10810 ax-resscn 10811 ax-1cn 10812 ax-icn 10813 ax-addcl 10814 ax-addrcl 10815 ax-mulcl 10816 ax-mulrcl 10817 ax-mulcom 10818 ax-addass 10819 ax-mulass 10820 ax-distr 10821 ax-i2m1 10822 ax-1ne0 10823 ax-1rid 10824 ax-rnegex 10825 ax-rrecex 10826 ax-cnre 10827 ax-pre-lttri 10828 ax-pre-lttrn 10829 ax-pre-ltadd 10830 ax-pre-mulgt0 10831 ax-addf 10833 ax-mulf 10834 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-int 4875 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-tr 5177 df-id 5470 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-se 5525 df-we 5526 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-pred 6176 df-ord 6234 df-on 6235 df-lim 6236 df-suc 6237 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-isom 6407 df-riota 7189 df-ov 7235 df-oprab 7236 df-mpo 7237 df-of 7488 df-om 7664 df-1st 7780 df-2nd 7781 df-supp 7925 df-wrecs 8068 df-recs 8129 df-rdg 8167 df-1o 8223 df-er 8412 df-map 8531 df-en 8648 df-dom 8649 df-sdom 8650 df-fin 8651 df-fsupp 9011 df-oi 9151 df-card 9580 df-pnf 10894 df-mnf 10895 df-xr 10896 df-ltxr 10897 df-le 10898 df-sub 11089 df-neg 11090 df-nn 11856 df-2 11918 df-3 11919 df-4 11920 df-5 11921 df-6 11922 df-7 11923 df-8 11924 df-9 11925 df-n0 12116 df-z 12202 df-dec 12319 df-uz 12464 df-fz 13121 df-fzo 13264 df-seq 13600 df-hash 13922 df-struct 16725 df-sets 16742 df-slot 16760 df-ndx 16770 df-base 16786 df-ress 16810 df-plusg 16840 df-mulr 16841 df-starv 16842 df-tset 16846 df-ple 16847 df-ds 16849 df-unif 16850 df-0g 16971 df-gsum 16972 df-mgm 18139 df-sgrp 18188 df-mnd 18199 df-submnd 18244 df-grp 18393 df-minusg 18394 df-cntz 18736 df-cmn 19197 df-abl 19198 df-mgp 19530 df-ur 19542 df-ring 19589 df-cring 19590 df-cnfld 20389 |
This theorem is referenced by: mdegmullem 25000 |
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