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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 21313 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
| 2 | cnfld0 21347 | . . 3 ⊢ 0 = (0g‘ℂfld) | |
| 3 | cnfldadd 21315 | . . 3 ⊢ + = (+g‘ℂfld) | |
| 4 | cnring 21345 | . . . 4 ⊢ ℂfld ∈ Ring | |
| 5 | ringcmn 20217 | . . . 4 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd) | |
| 6 | 4, 5 | mp1i 13 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ℂfld ∈ CMnd) |
| 7 | simpl 482 | . . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ 𝐴) | |
| 8 | tdeglem.a | . . . . . . . 8 ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} | |
| 9 | 8 | psrbagf 21874 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐴 → 𝑋:𝐼⟶ℕ0) |
| 10 | nn0sscn 12406 | . . . . . . 7 ⊢ ℕ0 ⊆ ℂ | |
| 11 | fss 6678 | . . . . . . 7 ⊢ ((𝑋:𝐼⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑋:𝐼⟶ℂ) | |
| 12 | 9, 10, 11 | sylancl 586 | . . . . . 6 ⊢ (𝑋 ∈ 𝐴 → 𝑋:𝐼⟶ℂ) |
| 13 | 12 | adantr 480 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋:𝐼⟶ℂ) |
| 14 | 13 | ffnd 6663 | . . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 Fn 𝐼) |
| 15 | 7, 14 | fndmexd 7846 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝐼 ∈ V) |
| 16 | 8 | psrbagf 21874 | . . . . 5 ⊢ (𝑌 ∈ 𝐴 → 𝑌:𝐼⟶ℕ0) |
| 17 | fss 6678 | . . . . 5 ⊢ ((𝑌:𝐼⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑌:𝐼⟶ℂ) | |
| 18 | 16, 10, 17 | sylancl 586 | . . . 4 ⊢ (𝑌 ∈ 𝐴 → 𝑌:𝐼⟶ℂ) |
| 19 | 18 | adantl 481 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌:𝐼⟶ℂ) |
| 20 | 8 | psrbagfsupp 21875 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → 𝑋 finSupp 0) |
| 21 | 20 | adantr 480 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 finSupp 0) |
| 22 | 8 | psrbagfsupp 21875 | . . . 4 ⊢ (𝑌 ∈ 𝐴 → 𝑌 finSupp 0) |
| 23 | 22 | adantl 481 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌 finSupp 0) |
| 24 | 1, 2, 3, 6, 15, 13, 19, 21, 23 | gsumadd 19852 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (ℂfld Σg (𝑋 ∘f + 𝑌)) = ((ℂfld Σg 𝑋) + (ℂfld Σg 𝑌))) |
| 25 | 8 | psrbagaddcl 21880 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∘f + 𝑌) ∈ 𝐴) |
| 26 | oveq2 7366 | . . . 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 7391 | . . . 4 ⊢ (ℂfld Σg (𝑋 ∘f + 𝑌)) ∈ V | |
| 29 | 26, 27, 28 | fvmpt 6941 | . . 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 7366 | . . . 4 ⊢ (ℎ = 𝑋 → (ℂfld Σg ℎ) = (ℂfld Σg 𝑋)) | |
| 32 | ovex 7391 | . . . 4 ⊢ (ℂfld Σg 𝑋) ∈ V | |
| 33 | 31, 27, 32 | fvmpt 6941 | . . 3 ⊢ (𝑋 ∈ 𝐴 → (𝐻‘𝑋) = (ℂfld Σg 𝑋)) |
| 34 | oveq2 7366 | . . . 4 ⊢ (ℎ = 𝑌 → (ℂfld Σg ℎ) = (ℂfld Σg 𝑌)) | |
| 35 | ovex 7391 | . . . 4 ⊢ (ℂfld Σg 𝑌) ∈ V | |
| 36 | 34, 27, 35 | fvmpt 6941 | . . 3 ⊢ (𝑌 ∈ 𝐴 → (𝐻‘𝑌) = (ℂfld Σg 𝑌)) |
| 37 | 33, 36 | oveqan12d 7377 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐻‘𝑋) + (𝐻‘𝑌)) = ((ℂfld Σg 𝑋) + (ℂfld Σg 𝑌))) |
| 38 | 24, 30, 37 | 3eqtr4d 2781 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐻‘(𝑋 ∘f + 𝑌)) = ((𝐻‘𝑋) + (𝐻‘𝑌))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 {crab 3399 Vcvv 3440 ⊆ wss 3901 class class class wbr 5098 ↦ cmpt 5179 ◡ccnv 5623 “ cima 5627 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ∘f cof 7620 ↑m cmap 8763 Fincfn 8883 finSupp cfsupp 9264 ℂcc 11024 0cc0 11026 + caddc 11029 ℕcn 12145 ℕ0cn0 12401 Σg cgsu 17360 CMndccmn 19709 Ringcrg 20168 ℂfldccnfld 21309 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-addf 11105 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-fz 13424 df-fzo 13571 df-seq 13925 df-hash 14254 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-0g 17361 df-gsum 17362 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-grp 18866 df-minusg 18867 df-cntz 19246 df-cmn 19711 df-abl 19712 df-mgp 20076 df-ur 20117 df-ring 20170 df-cring 20171 df-cnfld 21310 |
| This theorem is referenced by: mdegmullem 26039 |
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