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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.) |
Ref | Expression |
---|---|
tdeglem.a | ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} |
tdeglem.h | ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) |
Ref | Expression |
---|---|
tdeglem3 | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐻‘(𝑋 ∘f + 𝑌)) = ((𝐻‘𝑋) + (𝐻‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldbas 20552 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
2 | cnfld0 20572 | . . 3 ⊢ 0 = (0g‘ℂfld) | |
3 | cnfldadd 20553 | . . 3 ⊢ + = (+g‘ℂfld) | |
4 | cnring 20570 | . . . 4 ⊢ ℂfld ∈ Ring | |
5 | ringcmn 19334 | . . . 4 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd) | |
6 | 4, 5 | mp1i 13 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ℂfld ∈ CMnd) |
7 | simp1 1132 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝐼 ∈ 𝑉) | |
8 | tdeglem.a | . . . . . 6 ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} | |
9 | 8 | psrbagf 20148 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋:𝐼⟶ℕ0) |
10 | nn0sscn 11905 | . . . . 5 ⊢ ℕ0 ⊆ ℂ | |
11 | fss 6530 | . . . . 5 ⊢ ((𝑋:𝐼⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑋:𝐼⟶ℂ) | |
12 | 9, 10, 11 | sylancl 588 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋:𝐼⟶ℂ) |
13 | 12 | 3adant3 1128 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋:𝐼⟶ℂ) |
14 | 8 | psrbagf 20148 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝑌:𝐼⟶ℕ0) |
15 | fss 6530 | . . . . 5 ⊢ ((𝑌:𝐼⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑌:𝐼⟶ℂ) | |
16 | 14, 10, 15 | sylancl 588 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝑌:𝐼⟶ℂ) |
17 | 16 | 3adant2 1127 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌:𝐼⟶ℂ) |
18 | 8 | psrbagfsupp 20292 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐼 ∈ 𝑉) → 𝑋 finSupp 0) |
19 | 18 | ancoms 461 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋 finSupp 0) |
20 | 19 | 3adant3 1128 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 finSupp 0) |
21 | 8 | psrbagfsupp 20292 | . . . . 5 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐼 ∈ 𝑉) → 𝑌 finSupp 0) |
22 | 21 | ancoms 461 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝑌 finSupp 0) |
23 | 22 | 3adant2 1127 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌 finSupp 0) |
24 | 1, 2, 3, 6, 7, 13, 17, 20, 23 | gsumadd 19046 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (ℂfld Σg (𝑋 ∘f + 𝑌)) = ((ℂfld Σg 𝑋) + (ℂfld Σg 𝑌))) |
25 | 8 | psrbagaddcl 20153 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∘f + 𝑌) ∈ 𝐴) |
26 | oveq2 7167 | . . . 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 7192 | . . . 4 ⊢ (ℂfld Σg (𝑋 ∘f + 𝑌)) ∈ V | |
29 | 26, 27, 28 | fvmpt 6771 | . . 3 ⊢ ((𝑋 ∘f + 𝑌) ∈ 𝐴 → (𝐻‘(𝑋 ∘f + 𝑌)) = (ℂfld Σg (𝑋 ∘f + 𝑌))) |
30 | 25, 29 | syl 17 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐻‘(𝑋 ∘f + 𝑌)) = (ℂfld Σg (𝑋 ∘f + 𝑌))) |
31 | oveq2 7167 | . . . . 5 ⊢ (ℎ = 𝑋 → (ℂfld Σg ℎ) = (ℂfld Σg 𝑋)) | |
32 | ovex 7192 | . . . . 5 ⊢ (ℂfld Σg 𝑋) ∈ V | |
33 | 31, 27, 32 | fvmpt 6771 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝐻‘𝑋) = (ℂfld Σg 𝑋)) |
34 | oveq2 7167 | . . . . 5 ⊢ (ℎ = 𝑌 → (ℂfld Σg ℎ) = (ℂfld Σg 𝑌)) | |
35 | ovex 7192 | . . . . 5 ⊢ (ℂfld Σg 𝑌) ∈ V | |
36 | 34, 27, 35 | fvmpt 6771 | . . . 4 ⊢ (𝑌 ∈ 𝐴 → (𝐻‘𝑌) = (ℂfld Σg 𝑌)) |
37 | 33, 36 | oveqan12d 7178 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐻‘𝑋) + (𝐻‘𝑌)) = ((ℂfld Σg 𝑋) + (ℂfld Σg 𝑌))) |
38 | 37 | 3adant1 1126 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐻‘𝑋) + (𝐻‘𝑌)) = ((ℂfld Σg 𝑋) + (ℂfld Σg 𝑌))) |
39 | 24, 30, 38 | 3eqtr4d 2869 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐻‘(𝑋 ∘f + 𝑌)) = ((𝐻‘𝑋) + (𝐻‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 {crab 3145 ⊆ wss 3939 class class class wbr 5069 ↦ cmpt 5149 ◡ccnv 5557 “ cima 5561 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ∘f cof 7410 ↑m cmap 8409 Fincfn 8512 finSupp cfsupp 8836 ℂcc 10538 0cc0 10540 + caddc 10543 ℕcn 11641 ℕ0cn0 11900 Σg cgsu 16717 CMndccmn 18909 Ringcrg 19300 ℂfldccnfld 20548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-0g 16718 df-gsum 16719 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-grp 18109 df-minusg 18110 df-cntz 18450 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-cring 19303 df-cnfld 20549 |
This theorem is referenced by: mdegmullem 24675 |
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