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Mirrors > Home > MPE Home > Th. List > rrgeq0i | Structured version Visualization version GIF version |
Description: Property of a left-regular element. (Contributed by Stefan O'Rear, 22-Mar-2015.) |
Ref | Expression |
---|---|
rrgval.e | β’ πΈ = (RLRegβπ ) |
rrgval.b | β’ π΅ = (Baseβπ ) |
rrgval.t | β’ Β· = (.rβπ ) |
rrgval.z | β’ 0 = (0gβπ ) |
Ref | Expression |
---|---|
rrgeq0i | β’ ((π β πΈ β§ π β π΅) β ((π Β· π) = 0 β π = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrgval.e | . . . 4 β’ πΈ = (RLRegβπ ) | |
2 | rrgval.b | . . . 4 β’ π΅ = (Baseβπ ) | |
3 | rrgval.t | . . . 4 β’ Β· = (.rβπ ) | |
4 | rrgval.z | . . . 4 β’ 0 = (0gβπ ) | |
5 | 1, 2, 3, 4 | isrrg 21242 | . . 3 β’ (π β πΈ β (π β π΅ β§ βπ¦ β π΅ ((π Β· π¦) = 0 β π¦ = 0 ))) |
6 | 5 | simprbi 495 | . 2 β’ (π β πΈ β βπ¦ β π΅ ((π Β· π¦) = 0 β π¦ = 0 )) |
7 | oveq2 7434 | . . . . 5 β’ (π¦ = π β (π Β· π¦) = (π Β· π)) | |
8 | 7 | eqeq1d 2730 | . . . 4 β’ (π¦ = π β ((π Β· π¦) = 0 β (π Β· π) = 0 )) |
9 | eqeq1 2732 | . . . 4 β’ (π¦ = π β (π¦ = 0 β π = 0 )) | |
10 | 8, 9 | imbi12d 343 | . . 3 β’ (π¦ = π β (((π Β· π¦) = 0 β π¦ = 0 ) β ((π Β· π) = 0 β π = 0 ))) |
11 | 10 | rspcv 3607 | . 2 β’ (π β π΅ β (βπ¦ β π΅ ((π Β· π¦) = 0 β π¦ = 0 ) β ((π Β· π) = 0 β π = 0 ))) |
12 | 6, 11 | mpan9 505 | 1 β’ ((π β πΈ β§ π β π΅) β ((π Β· π) = 0 β π = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3058 βcfv 6553 (class class class)co 7426 Basecbs 17187 .rcmulr 17241 0gc0g 17428 RLRegcrlreg 21233 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7429 df-rlreg 21237 |
This theorem is referenced by: rrgeq0 21244 znrrg 21506 deg1mul2 26070 rrgsubm 32976 rlocf1 33012 fracerl 33017 |
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