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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lfladdcom | Structured version Visualization version GIF version |
Description: Commutativity of functional addition. (Contributed by NM, 19-Oct-2014.) |
Ref | Expression |
---|---|
lfladdcl.r | β’ π = (Scalarβπ) |
lfladdcl.p | β’ + = (+gβπ ) |
lfladdcl.f | β’ πΉ = (LFnlβπ) |
lfladdcl.w | β’ (π β π β LMod) |
lfladdcl.g | β’ (π β πΊ β πΉ) |
lfladdcl.h | β’ (π β π» β πΉ) |
Ref | Expression |
---|---|
lfladdcom | β’ (π β (πΊ βf + π») = (π» βf + πΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvexd 6907 | . 2 β’ (π β (Baseβπ) β V) | |
2 | lfladdcl.w | . . 3 β’ (π β π β LMod) | |
3 | lfladdcl.g | . . 3 β’ (π β πΊ β πΉ) | |
4 | lfladdcl.r | . . . 4 β’ π = (Scalarβπ) | |
5 | eqid 2725 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
6 | eqid 2725 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
7 | lfladdcl.f | . . . 4 β’ πΉ = (LFnlβπ) | |
8 | 4, 5, 6, 7 | lflf 38591 | . . 3 β’ ((π β LMod β§ πΊ β πΉ) β πΊ:(Baseβπ)βΆ(Baseβπ )) |
9 | 2, 3, 8 | syl2anc 582 | . 2 β’ (π β πΊ:(Baseβπ)βΆ(Baseβπ )) |
10 | lfladdcl.h | . . 3 β’ (π β π» β πΉ) | |
11 | 4, 5, 6, 7 | lflf 38591 | . . 3 β’ ((π β LMod β§ π» β πΉ) β π»:(Baseβπ)βΆ(Baseβπ )) |
12 | 2, 10, 11 | syl2anc 582 | . 2 β’ (π β π»:(Baseβπ)βΆ(Baseβπ )) |
13 | 4 | lmodring 20755 | . . . . 5 β’ (π β LMod β π β Ring) |
14 | ringabl 20221 | . . . . 5 β’ (π β Ring β π β Abel) | |
15 | 2, 13, 14 | 3syl 18 | . . . 4 β’ (π β π β Abel) |
16 | 15 | adantr 479 | . . 3 β’ ((π β§ (π₯ β (Baseβπ ) β§ π¦ β (Baseβπ ))) β π β Abel) |
17 | simprl 769 | . . 3 β’ ((π β§ (π₯ β (Baseβπ ) β§ π¦ β (Baseβπ ))) β π₯ β (Baseβπ )) | |
18 | simprr 771 | . . 3 β’ ((π β§ (π₯ β (Baseβπ ) β§ π¦ β (Baseβπ ))) β π¦ β (Baseβπ )) | |
19 | lfladdcl.p | . . . 4 β’ + = (+gβπ ) | |
20 | 5, 19 | ablcom 19758 | . . 3 β’ ((π β Abel β§ π₯ β (Baseβπ ) β§ π¦ β (Baseβπ )) β (π₯ + π¦) = (π¦ + π₯)) |
21 | 16, 17, 18, 20 | syl3anc 1368 | . 2 β’ ((π β§ (π₯ β (Baseβπ ) β§ π¦ β (Baseβπ ))) β (π₯ + π¦) = (π¦ + π₯)) |
22 | 1, 9, 12, 21 | caofcom 7718 | 1 β’ (π β (πΊ βf + π») = (π» βf + πΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3463 βΆwf 6539 βcfv 6543 (class class class)co 7416 βf cof 7680 Basecbs 17179 +gcplusg 17232 Scalarcsca 17235 Abelcabl 19740 Ringcrg 20177 LModclmod 20747 LFnlclfn 38585 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-cmn 19741 df-abl 19742 df-mgp 20079 df-ur 20126 df-ring 20179 df-lmod 20749 df-lfl 38586 |
This theorem is referenced by: ldualvaddcom 38668 |
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