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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 20800 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
2 | cnfld0 20821 | . . 3 ⊢ 0 = (0g‘ℂfld) | |
3 | cnfldadd 20801 | . . 3 ⊢ + = (+g‘ℂfld) | |
4 | cnring 20819 | . . . 4 ⊢ ℂfld ∈ Ring | |
5 | ringcmn 20003 | . . . 4 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd) | |
6 | 4, 5 | mp1i 13 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ℂfld ∈ CMnd) |
7 | simpl 483 | . . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ 𝐴) | |
8 | tdeglem.a | . . . . . . . 8 ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} | |
9 | 8 | psrbagf 21320 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐴 → 𝑋:𝐼⟶ℕ0) |
10 | nn0sscn 12418 | . . . . . . 7 ⊢ ℕ0 ⊆ ℂ | |
11 | fss 6685 | . . . . . . 7 ⊢ ((𝑋:𝐼⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑋:𝐼⟶ℂ) | |
12 | 9, 10, 11 | sylancl 586 | . . . . . 6 ⊢ (𝑋 ∈ 𝐴 → 𝑋:𝐼⟶ℂ) |
13 | 12 | adantr 481 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋:𝐼⟶ℂ) |
14 | 13 | ffnd 6669 | . . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 Fn 𝐼) |
15 | 7, 14 | fndmexd 7843 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝐼 ∈ V) |
16 | 8 | psrbagf 21320 | . . . . 5 ⊢ (𝑌 ∈ 𝐴 → 𝑌:𝐼⟶ℕ0) |
17 | fss 6685 | . . . . 5 ⊢ ((𝑌:𝐼⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑌:𝐼⟶ℂ) | |
18 | 16, 10, 17 | sylancl 586 | . . . 4 ⊢ (𝑌 ∈ 𝐴 → 𝑌:𝐼⟶ℂ) |
19 | 18 | adantl 482 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌:𝐼⟶ℂ) |
20 | 8 | psrbagfsupp 21322 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → 𝑋 finSupp 0) |
21 | 20 | adantr 481 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 finSupp 0) |
22 | 8 | psrbagfsupp 21322 | . . . 4 ⊢ (𝑌 ∈ 𝐴 → 𝑌 finSupp 0) |
23 | 22 | adantl 482 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌 finSupp 0) |
24 | 1, 2, 3, 6, 15, 13, 19, 21, 23 | gsumadd 19700 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (ℂfld Σg (𝑋 ∘f + 𝑌)) = ((ℂfld Σg 𝑋) + (ℂfld Σg 𝑌))) |
25 | 8 | psrbagaddcl 21330 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∘f + 𝑌) ∈ 𝐴) |
26 | oveq2 7365 | . . . 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 7390 | . . . 4 ⊢ (ℂfld Σg (𝑋 ∘f + 𝑌)) ∈ V | |
29 | 26, 27, 28 | fvmpt 6948 | . . 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 7365 | . . . 4 ⊢ (ℎ = 𝑋 → (ℂfld Σg ℎ) = (ℂfld Σg 𝑋)) | |
32 | ovex 7390 | . . . 4 ⊢ (ℂfld Σg 𝑋) ∈ V | |
33 | 31, 27, 32 | fvmpt 6948 | . . 3 ⊢ (𝑋 ∈ 𝐴 → (𝐻‘𝑋) = (ℂfld Σg 𝑋)) |
34 | oveq2 7365 | . . . 4 ⊢ (ℎ = 𝑌 → (ℂfld Σg ℎ) = (ℂfld Σg 𝑌)) | |
35 | ovex 7390 | . . . 4 ⊢ (ℂfld Σg 𝑌) ∈ V | |
36 | 34, 27, 35 | fvmpt 6948 | . . 3 ⊢ (𝑌 ∈ 𝐴 → (𝐻‘𝑌) = (ℂfld Σg 𝑌)) |
37 | 33, 36 | oveqan12d 7376 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐻‘𝑋) + (𝐻‘𝑌)) = ((ℂfld Σg 𝑋) + (ℂfld Σg 𝑌))) |
38 | 24, 30, 37 | 3eqtr4d 2786 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐻‘(𝑋 ∘f + 𝑌)) = ((𝐻‘𝑋) + (𝐻‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3407 Vcvv 3445 ⊆ wss 3910 class class class wbr 5105 ↦ cmpt 5188 ◡ccnv 5632 “ cima 5636 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ∘f cof 7615 ↑m cmap 8765 Fincfn 8883 finSupp cfsupp 9305 ℂcc 11049 0cc0 11051 + caddc 11054 ℕcn 12153 ℕ0cn0 12413 Σg cgsu 17322 CMndccmn 19562 Ringcrg 19964 ℂfldccnfld 20796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-0g 17323 df-gsum 17324 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-cnfld 20797 |
This theorem is referenced by: mdegmullem 25443 |
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