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Mirrors > Home > MPE Home > Th. List > tdeglem3OLD | Structured version Visualization version GIF version |
Description: Obsolete version of tdeglem3 25557 as of 7-Aug-2024. (Contributed by Stefan O'Rear, 26-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 |
---|---|
tdeglem3OLD | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐻‘(𝑋 ∘f + 𝑌)) = ((𝐻‘𝑋) + (𝐻‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldbas 20933 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
2 | cnfld0 20954 | . . 3 ⊢ 0 = (0g‘ℂfld) | |
3 | cnfldadd 20934 | . . 3 ⊢ + = (+g‘ℂfld) | |
4 | cnring 20952 | . . . 4 ⊢ ℂfld ∈ Ring | |
5 | ringcmn 20089 | . . . 4 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd) | |
6 | 4, 5 | mp1i 13 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ℂfld ∈ CMnd) |
7 | simp1 1137 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝐼 ∈ 𝑉) | |
8 | tdeglem.a | . . . . . 6 ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} | |
9 | 8 | psrbagfOLD 21454 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋:𝐼⟶ℕ0) |
10 | nn0sscn 12473 | . . . . 5 ⊢ ℕ0 ⊆ ℂ | |
11 | fss 6731 | . . . . 5 ⊢ ((𝑋:𝐼⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑋:𝐼⟶ℂ) | |
12 | 9, 10, 11 | sylancl 587 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋:𝐼⟶ℂ) |
13 | 12 | 3adant3 1133 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋:𝐼⟶ℂ) |
14 | 8 | psrbagfOLD 21454 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝑌:𝐼⟶ℕ0) |
15 | fss 6731 | . . . . 5 ⊢ ((𝑌:𝐼⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑌:𝐼⟶ℂ) | |
16 | 14, 10, 15 | sylancl 587 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝑌:𝐼⟶ℂ) |
17 | 16 | 3adant2 1132 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌:𝐼⟶ℂ) |
18 | 8 | psrbagfsuppOLD 21456 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐼 ∈ 𝑉) → 𝑋 finSupp 0) |
19 | 18 | ancoms 460 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋 finSupp 0) |
20 | 19 | 3adant3 1133 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 finSupp 0) |
21 | 8 | psrbagfsuppOLD 21456 | . . . . 5 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐼 ∈ 𝑉) → 𝑌 finSupp 0) |
22 | 21 | ancoms 460 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝑌 finSupp 0) |
23 | 22 | 3adant2 1132 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌 finSupp 0) |
24 | 1, 2, 3, 6, 7, 13, 17, 20, 23 | gsumadd 19783 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (ℂfld Σg (𝑋 ∘f + 𝑌)) = ((ℂfld Σg 𝑋) + (ℂfld Σg 𝑌))) |
25 | 8 | psrbagaddclOLD 21464 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∘f + 𝑌) ∈ 𝐴) |
26 | oveq2 7412 | . . . 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 7437 | . . . 4 ⊢ (ℂfld Σg (𝑋 ∘f + 𝑌)) ∈ V | |
29 | 26, 27, 28 | fvmpt 6994 | . . 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 7412 | . . . . 5 ⊢ (ℎ = 𝑋 → (ℂfld Σg ℎ) = (ℂfld Σg 𝑋)) | |
32 | ovex 7437 | . . . . 5 ⊢ (ℂfld Σg 𝑋) ∈ V | |
33 | 31, 27, 32 | fvmpt 6994 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝐻‘𝑋) = (ℂfld Σg 𝑋)) |
34 | oveq2 7412 | . . . . 5 ⊢ (ℎ = 𝑌 → (ℂfld Σg ℎ) = (ℂfld Σg 𝑌)) | |
35 | ovex 7437 | . . . . 5 ⊢ (ℂfld Σg 𝑌) ∈ V | |
36 | 34, 27, 35 | fvmpt 6994 | . . . 4 ⊢ (𝑌 ∈ 𝐴 → (𝐻‘𝑌) = (ℂfld Σg 𝑌)) |
37 | 33, 36 | oveqan12d 7423 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐻‘𝑋) + (𝐻‘𝑌)) = ((ℂfld Σg 𝑋) + (ℂfld Σg 𝑌))) |
38 | 37 | 3adant1 1131 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐻‘𝑋) + (𝐻‘𝑌)) = ((ℂfld Σg 𝑋) + (ℂfld Σg 𝑌))) |
39 | 24, 30, 38 | 3eqtr4d 2783 | 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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {crab 3433 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 ◡ccnv 5674 “ cima 5678 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 ∘f cof 7663 ↑m cmap 8816 Fincfn 8935 finSupp cfsupp 9357 ℂcc 11104 0cc0 11106 + caddc 11109 ℕcn 12208 ℕ0cn0 12468 Σg cgsu 17382 CMndccmn 19641 Ringcrg 20047 ℂfldccnfld 20929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-0g 17383 df-gsum 17384 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-cntz 19175 df-cmn 19643 df-abl 19644 df-mgp 19980 df-ur 19997 df-ring 20049 df-cring 20050 df-cnfld 20930 |
This theorem is referenced by: (None) |
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