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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 | β’ π΄ = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} |
tdeglem.h | β’ π» = (β β π΄ β¦ (βfld Ξ£g β)) |
Ref | Expression |
---|---|
tdeglem3 | β’ ((π β π΄ β§ π β π΄) β (π»β(π βf + π)) = ((π»βπ) + (π»βπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldbas 21238 | . . 3 β’ β = (Baseββfld) | |
2 | cnfld0 21274 | . . 3 β’ 0 = (0gββfld) | |
3 | cnfldadd 21240 | . . 3 β’ + = (+gββfld) | |
4 | cnring 21272 | . . . 4 β’ βfld β Ring | |
5 | ringcmn 20177 | . . . 4 β’ (βfld β Ring β βfld β CMnd) | |
6 | 4, 5 | mp1i 13 | . . 3 β’ ((π β π΄ β§ π β π΄) β βfld β CMnd) |
7 | simpl 482 | . . . 4 β’ ((π β π΄ β§ π β π΄) β π β π΄) | |
8 | tdeglem.a | . . . . . . . 8 β’ π΄ = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} | |
9 | 8 | psrbagf 21801 | . . . . . . 7 β’ (π β π΄ β π:πΌβΆβ0) |
10 | nn0sscn 12476 | . . . . . . 7 β’ β0 β β | |
11 | fss 6725 | . . . . . . 7 β’ ((π:πΌβΆβ0 β§ β0 β β) β π:πΌβΆβ) | |
12 | 9, 10, 11 | sylancl 585 | . . . . . 6 β’ (π β π΄ β π:πΌβΆβ) |
13 | 12 | adantr 480 | . . . . 5 β’ ((π β π΄ β§ π β π΄) β π:πΌβΆβ) |
14 | 13 | ffnd 6709 | . . . 4 β’ ((π β π΄ β§ π β π΄) β π Fn πΌ) |
15 | 7, 14 | fndmexd 7891 | . . 3 β’ ((π β π΄ β§ π β π΄) β πΌ β V) |
16 | 8 | psrbagf 21801 | . . . . 5 β’ (π β π΄ β π:πΌβΆβ0) |
17 | fss 6725 | . . . . 5 β’ ((π:πΌβΆβ0 β§ β0 β β) β π:πΌβΆβ) | |
18 | 16, 10, 17 | sylancl 585 | . . . 4 β’ (π β π΄ β π:πΌβΆβ) |
19 | 18 | adantl 481 | . . 3 β’ ((π β π΄ β§ π β π΄) β π:πΌβΆβ) |
20 | 8 | psrbagfsupp 21803 | . . . 4 β’ (π β π΄ β π finSupp 0) |
21 | 20 | adantr 480 | . . 3 β’ ((π β π΄ β§ π β π΄) β π finSupp 0) |
22 | 8 | psrbagfsupp 21803 | . . . 4 β’ (π β π΄ β π finSupp 0) |
23 | 22 | adantl 481 | . . 3 β’ ((π β π΄ β§ π β π΄) β π finSupp 0) |
24 | 1, 2, 3, 6, 15, 13, 19, 21, 23 | gsumadd 19839 | . 2 β’ ((π β π΄ β§ π β π΄) β (βfld Ξ£g (π βf + π)) = ((βfld Ξ£g π) + (βfld Ξ£g π))) |
25 | 8 | psrbagaddcl 21811 | . . 3 β’ ((π β π΄ β§ π β π΄) β (π βf + π) β π΄) |
26 | oveq2 7410 | . . . 4 β’ (β = (π βf + π) β (βfld Ξ£g β) = (βfld Ξ£g (π βf + π))) | |
27 | tdeglem.h | . . . 4 β’ π» = (β β π΄ β¦ (βfld Ξ£g β)) | |
28 | ovex 7435 | . . . 4 β’ (βfld Ξ£g (π βf + π)) β V | |
29 | 26, 27, 28 | fvmpt 6989 | . . 3 β’ ((π βf + π) β π΄ β (π»β(π βf + π)) = (βfld Ξ£g (π βf + π))) |
30 | 25, 29 | syl 17 | . 2 β’ ((π β π΄ β§ π β π΄) β (π»β(π βf + π)) = (βfld Ξ£g (π βf + π))) |
31 | oveq2 7410 | . . . 4 β’ (β = π β (βfld Ξ£g β) = (βfld Ξ£g π)) | |
32 | ovex 7435 | . . . 4 β’ (βfld Ξ£g π) β V | |
33 | 31, 27, 32 | fvmpt 6989 | . . 3 β’ (π β π΄ β (π»βπ) = (βfld Ξ£g π)) |
34 | oveq2 7410 | . . . 4 β’ (β = π β (βfld Ξ£g β) = (βfld Ξ£g π)) | |
35 | ovex 7435 | . . . 4 β’ (βfld Ξ£g π) β V | |
36 | 34, 27, 35 | fvmpt 6989 | . . 3 β’ (π β π΄ β (π»βπ) = (βfld Ξ£g π)) |
37 | 33, 36 | oveqan12d 7421 | . 2 β’ ((π β π΄ β§ π β π΄) β ((π»βπ) + (π»βπ)) = ((βfld Ξ£g π) + (βfld Ξ£g π))) |
38 | 24, 30, 37 | 3eqtr4d 2774 | 1 β’ ((π β π΄ β§ π β π΄) β (π»β(π βf + π)) = ((π»βπ) + (π»βπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {crab 3424 Vcvv 3466 β wss 3941 class class class wbr 5139 β¦ cmpt 5222 β‘ccnv 5666 β cima 5670 βΆwf 6530 βcfv 6534 (class class class)co 7402 βf cof 7662 βm cmap 8817 Fincfn 8936 finSupp cfsupp 9358 βcc 11105 0cc0 11107 + caddc 11110 βcn 12211 β0cn0 12471 Ξ£g cgsu 17391 CMndccmn 19696 Ringcrg 20134 βfldccnfld 21234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-addf 11186 ax-mulf 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-fzo 13629 df-seq 13968 df-hash 14292 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-0g 17392 df-gsum 17393 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-grp 18862 df-minusg 18863 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-ur 20083 df-ring 20136 df-cring 20137 df-cnfld 21235 |
This theorem is referenced by: mdegmullem 25958 |
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