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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > evpmval | Structured version Visualization version GIF version |
Description: Value of the set of even permutations, the alternating group. (Contributed by Thierry Arnoux, 1-Nov-2023.) |
Ref | Expression |
---|---|
evpmval.1 | β’ π΄ = (pmEvenβπ·) |
Ref | Expression |
---|---|
evpmval | β’ (π· β π β π΄ = (β‘(pmSgnβπ·) β {1})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evpmval.1 | . 2 β’ π΄ = (pmEvenβπ·) | |
2 | elex 3462 | . . 3 β’ (π· β π β π· β V) | |
3 | fveq2 6843 | . . . . . 6 β’ (π = π· β (pmSgnβπ) = (pmSgnβπ·)) | |
4 | 3 | cnveqd 5832 | . . . . 5 β’ (π = π· β β‘(pmSgnβπ) = β‘(pmSgnβπ·)) |
5 | 4 | imaeq1d 6013 | . . . 4 β’ (π = π· β (β‘(pmSgnβπ) β {1}) = (β‘(pmSgnβπ·) β {1})) |
6 | df-evpm 19279 | . . . 4 β’ pmEven = (π β V β¦ (β‘(pmSgnβπ) β {1})) | |
7 | fvex 6856 | . . . . . 6 β’ (pmSgnβπ·) β V | |
8 | 7 | cnvex 7863 | . . . . 5 β’ β‘(pmSgnβπ·) β V |
9 | 8 | imaex 7854 | . . . 4 β’ (β‘(pmSgnβπ·) β {1}) β V |
10 | 5, 6, 9 | fvmpt 6949 | . . 3 β’ (π· β V β (pmEvenβπ·) = (β‘(pmSgnβπ·) β {1})) |
11 | 2, 10 | syl 17 | . 2 β’ (π· β π β (pmEvenβπ·) = (β‘(pmSgnβπ·) β {1})) |
12 | 1, 11 | eqtrid 2785 | 1 β’ (π· β π β π΄ = (β‘(pmSgnβπ·) β {1})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3444 {csn 4587 β‘ccnv 5633 β cima 5637 βcfv 6497 1c1 11057 pmSgncpsgn 19276 pmEvencevpm 19277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fv 6505 df-evpm 19279 |
This theorem is referenced by: evpmsubg 32045 |
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