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Mirrors > Home > MPE Home > Th. List > tdeglem1OLD | Structured version Visualization version GIF version |
Description: Obsolete version of tdeglem1 24769 as of 7-Aug-2024. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
tdeglem.a | ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} |
tdeglem.h | ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) |
Ref | Expression |
---|---|
tdeglem1OLD | ⊢ (𝐼 ∈ 𝑉 → 𝐻:𝐴⟶ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfld0 20204 | . . 3 ⊢ 0 = (0g‘ℂfld) | |
2 | cnring 20202 | . . . 4 ⊢ ℂfld ∈ Ring | |
3 | ringcmn 19416 | . . . 4 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd) | |
4 | 2, 3 | mp1i 13 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ ℎ ∈ 𝐴) → ℂfld ∈ CMnd) |
5 | simpl 486 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ ℎ ∈ 𝐴) → 𝐼 ∈ 𝑉) | |
6 | nn0subm 20235 | . . . 4 ⊢ ℕ0 ∈ (SubMnd‘ℂfld) | |
7 | 6 | a1i 11 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ ℎ ∈ 𝐴) → ℕ0 ∈ (SubMnd‘ℂfld)) |
8 | tdeglem.a | . . . 4 ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} | |
9 | 8 | psrbagfOLD 20695 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ ℎ ∈ 𝐴) → ℎ:𝐼⟶ℕ0) |
10 | 8 | psrbagfsuppOLD 20697 | . . . 4 ⊢ ((ℎ ∈ 𝐴 ∧ 𝐼 ∈ 𝑉) → ℎ finSupp 0) |
11 | 10 | ancoms 462 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ ℎ ∈ 𝐴) → ℎ finSupp 0) |
12 | 1, 4, 5, 7, 9, 11 | gsumsubmcl 19121 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ ℎ ∈ 𝐴) → (ℂfld Σg ℎ) ∈ ℕ0) |
13 | tdeglem.h | . 2 ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) | |
14 | 12, 13 | fmptd 6875 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐻:𝐴⟶ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3074 class class class wbr 5036 ↦ cmpt 5116 ◡ccnv 5527 “ cima 5531 ⟶wf 6336 ‘cfv 6340 (class class class)co 7156 ↑m cmap 8422 Fincfn 8540 finSupp cfsupp 8879 0cc0 10588 ℕcn 11687 ℕ0cn0 11947 Σg cgsu 16786 SubMndcsubmnd 18035 CMndccmn 18987 Ringcrg 19379 ℂfldccnfld 20180 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 ax-addf 10667 ax-mulf 10668 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-se 5488 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7586 df-1st 7699 df-2nd 7700 df-supp 7842 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-1o 8118 df-er 8305 df-map 8424 df-en 8541 df-dom 8542 df-sdom 8543 df-fin 8544 df-fsupp 8880 df-oi 9020 df-card 9414 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-nn 11688 df-2 11750 df-3 11751 df-4 11752 df-5 11753 df-6 11754 df-7 11755 df-8 11756 df-9 11757 df-n0 11948 df-z 12034 df-dec 12151 df-uz 12296 df-fz 12953 df-fzo 13096 df-seq 13432 df-hash 13754 df-struct 16557 df-ndx 16558 df-slot 16559 df-base 16561 df-sets 16562 df-ress 16563 df-plusg 16650 df-mulr 16651 df-starv 16652 df-tset 16656 df-ple 16657 df-ds 16659 df-unif 16660 df-0g 16787 df-gsum 16788 df-mgm 17932 df-sgrp 17981 df-mnd 17992 df-submnd 18037 df-grp 18186 df-minusg 18187 df-cntz 18528 df-cmn 18989 df-abl 18990 df-mgp 19322 df-ur 19334 df-ring 19381 df-cring 19382 df-cnfld 20181 |
This theorem is referenced by: (None) |
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