Description: Nonnegative multiple of a group sum. (Contributed by Mario Carneiro, 15-Dec-2014) (Revised by Mario Carneiro, 7-Jan-2015) (Revised by AV, 6-Jun-2019)
Ref | Expression | ||
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Hypotheses | gsummulg.b | |- B = ( Base ` G ) |
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gsummulg.z | |- .0. = ( 0g ` G ) |
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gsummulg.t | |- .x. = ( .g ` G ) |
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gsummulg.a | |- ( ph -> A e. V ) |
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gsummulg.f | |- ( ( ph /\ k e. A ) -> X e. B ) |
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gsummulg.w | |- ( ph -> ( k e. A |-> X ) finSupp .0. ) |
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gsummulg.g | |- ( ph -> G e. CMnd ) |
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gsummulg.n | |- ( ph -> N e. NN0 ) |
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Assertion | gsummulg | |- ( ph -> ( G gsum ( k e. A |-> ( N .x. X ) ) ) = ( N .x. ( G gsum ( k e. A |-> X ) ) ) ) |
Step | Hyp | Ref | Expression |
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1 | gsummulg.b | |- B = ( Base ` G ) |
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2 | gsummulg.z | |- .0. = ( 0g ` G ) |
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3 | gsummulg.t | |- .x. = ( .g ` G ) |
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4 | gsummulg.a | |- ( ph -> A e. V ) |
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5 | gsummulg.f | |- ( ( ph /\ k e. A ) -> X e. B ) |
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6 | gsummulg.w | |- ( ph -> ( k e. A |-> X ) finSupp .0. ) |
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7 | gsummulg.g | |- ( ph -> G e. CMnd ) |
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8 | gsummulg.n | |- ( ph -> N e. NN0 ) |
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9 | 8 | nn0zd | |- ( ph -> N e. ZZ ) |
10 | 8 | olcd | |- ( ph -> ( G e. Abel \/ N e. NN0 ) ) |
11 | 1 2 3 4 5 6 7 9 10 | gsummulglem | |- ( ph -> ( G gsum ( k e. A |-> ( N .x. X ) ) ) = ( N .x. ( G gsum ( k e. A |-> X ) ) ) ) |