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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 21283 | . . 3 β’ β = (Baseββfld) | |
2 | cnfld0 21320 | . . 3 β’ 0 = (0gββfld) | |
3 | cnfldadd 21285 | . . 3 β’ + = (+gββfld) | |
4 | cnring 21318 | . . . 4 β’ βfld β Ring | |
5 | ringcmn 20218 | . . . 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 21851 | . . . . . . 7 β’ (π β π΄ β π:πΌβΆβ0) |
10 | nn0sscn 12508 | . . . . . . 7 β’ β0 β β | |
11 | fss 6739 | . . . . . . 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 6723 | . . . 4 β’ ((π β π΄ β§ π β π΄) β π Fn πΌ) |
15 | 7, 14 | fndmexd 7912 | . . 3 β’ ((π β π΄ β§ π β π΄) β πΌ β V) |
16 | 8 | psrbagf 21851 | . . . . 5 β’ (π β π΄ β π:πΌβΆβ0) |
17 | fss 6739 | . . . . 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 21853 | . . . 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 21853 | . . . 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 19878 | . 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 21861 | . . 3 β’ ((π β π΄ β§ π β π΄) β (π βf + π) β π΄) |
26 | oveq2 7428 | . . . 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 7453 | . . . 4 β’ (βfld Ξ£g (π βf + π)) β V | |
29 | 26, 27, 28 | fvmpt 7005 | . . 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 7428 | . . . 4 β’ (β = π β (βfld Ξ£g β) = (βfld Ξ£g π)) | |
32 | ovex 7453 | . . . 4 β’ (βfld Ξ£g π) β V | |
33 | 31, 27, 32 | fvmpt 7005 | . . 3 β’ (π β π΄ β (π»βπ) = (βfld Ξ£g π)) |
34 | oveq2 7428 | . . . 4 β’ (β = π β (βfld Ξ£g β) = (βfld Ξ£g π)) | |
35 | ovex 7453 | . . . 4 β’ (βfld Ξ£g π) β V | |
36 | 34, 27, 35 | fvmpt 7005 | . . 3 β’ (π β π΄ β (π»βπ) = (βfld Ξ£g π)) |
37 | 33, 36 | oveqan12d 7439 | . 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 2778 | 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 1534 β wcel 2099 {crab 3429 Vcvv 3471 β wss 3947 class class class wbr 5148 β¦ cmpt 5231 β‘ccnv 5677 β cima 5681 βΆwf 6544 βcfv 6548 (class class class)co 7420 βf cof 7683 βm cmap 8845 Fincfn 8964 finSupp cfsupp 9386 βcc 11137 0cc0 11139 + caddc 11142 βcn 12243 β0cn0 12503 Ξ£g cgsu 17422 CMndccmn 19735 Ringcrg 20173 βfldccnfld 21279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-addf 11218 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-0g 17423 df-gsum 17424 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-grp 18893 df-minusg 18894 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-ur 20122 df-ring 20175 df-cring 20176 df-cnfld 21280 |
This theorem is referenced by: mdegmullem 26027 |
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