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Mirrors > Home > ILE Home > Th. List > mgpvalg | GIF version |
Description: Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.) |
Ref | Expression |
---|---|
mgpval.1 | β’ π = (mulGrpβπ ) |
mgpval.2 | β’ Β· = (.rβπ ) |
Ref | Expression |
---|---|
mgpvalg | β’ (π β π β π = (π sSet β¨(+gβndx), Β· β©)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgpval.1 | . 2 β’ π = (mulGrpβπ ) | |
2 | df-mgp 13147 | . . 3 β’ mulGrp = (π β V β¦ (π sSet β¨(+gβndx), (.rβπ)β©)) | |
3 | id 19 | . . . 4 β’ (π = π β π = π ) | |
4 | fveq2 5517 | . . . . . 6 β’ (π = π β (.rβπ) = (.rβπ )) | |
5 | mgpval.2 | . . . . . 6 β’ Β· = (.rβπ ) | |
6 | 4, 5 | eqtr4di 2228 | . . . . 5 β’ (π = π β (.rβπ) = Β· ) |
7 | 6 | opeq2d 3787 | . . . 4 β’ (π = π β β¨(+gβndx), (.rβπ)β© = β¨(+gβndx), Β· β©) |
8 | 3, 7 | oveq12d 5896 | . . 3 β’ (π = π β (π sSet β¨(+gβndx), (.rβπ)β©) = (π sSet β¨(+gβndx), Β· β©)) |
9 | elex 2750 | . . 3 β’ (π β π β π β V) | |
10 | plusgslid 12574 | . . . . . 6 β’ (+g = Slot (+gβndx) β§ (+gβndx) β β) | |
11 | 10 | simpri 113 | . . . . 5 β’ (+gβndx) β β |
12 | 11 | a1i 9 | . . . 4 β’ (π β π β (+gβndx) β β) |
13 | mulrslid 12593 | . . . . . 6 β’ (.r = Slot (.rβndx) β§ (.rβndx) β β) | |
14 | 13 | slotex 12492 | . . . . 5 β’ (π β π β (.rβπ ) β V) |
15 | 5, 14 | eqeltrid 2264 | . . . 4 β’ (π β π β Β· β V) |
16 | setsex 12497 | . . . 4 β’ ((π β π β§ (+gβndx) β β β§ Β· β V) β (π sSet β¨(+gβndx), Β· β©) β V) | |
17 | 12, 15, 16 | mpd3an23 1339 | . . 3 β’ (π β π β (π sSet β¨(+gβndx), Β· β©) β V) |
18 | 2, 8, 9, 17 | fvmptd3 5612 | . 2 β’ (π β π β (mulGrpβπ ) = (π sSet β¨(+gβndx), Β· β©)) |
19 | 1, 18 | eqtrid 2222 | 1 β’ (π β π β π = (π sSet β¨(+gβndx), Β· β©)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1353 β wcel 2148 Vcvv 2739 β¨cop 3597 βcfv 5218 (class class class)co 5878 βcn 8922 ndxcnx 12462 sSet csts 12463 Slot cslot 12464 +gcplusg 12539 .rcmulr 12540 mulGrpcmgp 13146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1re 7908 ax-addrcl 7911 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-iota 5180 df-fun 5220 df-fn 5221 df-fv 5226 df-ov 5881 df-oprab 5882 df-mpo 5883 df-inn 8923 df-2 8981 df-3 8982 df-ndx 12468 df-slot 12469 df-sets 12472 df-plusg 12552 df-mulr 12553 df-mgp 13147 |
This theorem is referenced by: mgpplusgg 13150 mgpex 13151 mgpbasg 13152 mgpscag 13153 mgptsetg 13154 mgpdsg 13156 mgpress 13157 |
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