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Mirrors > Home > ILE Home > Th. List > add32d | Unicode version |
Description: Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
addd.1 |
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addd.2 |
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addd.3 |
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Ref | Expression |
---|---|
add32d |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addd.1 |
. 2
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2 | addd.2 |
. 2
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3 | addd.3 |
. 2
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4 | add32 7945 |
. 2
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5 | 1, 2, 3, 4 | syl3anc 1217 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-addcom 7744 ax-addass 7746 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785 |
This theorem is referenced by: nppcan 8008 muladd 8170 peano5uzti 9183 flqaddz 10101 seq3shft2 10277 zesq 10441 abstri 10908 bdtrilem 11042 |
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