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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 | |
addd.2 | |
addd.3 |
Ref | Expression |
---|---|
add32d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addd.1 | . 2 | |
2 | addd.2 | . 2 | |
3 | addd.3 | . 2 | |
4 | add32 7170 | . 2 | |
5 | 1, 2, 3, 4 | syl3anc 1135 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1243 wcel 1393 (class class class)co 5512 cc 6887 caddc 6892 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-addcom 6984 ax-addass 6986 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 |
This theorem is referenced by: nppcan 7233 muladd 7381 peano5uzti 8346 flqaddz 9139 zesq 9367 abstri 9700 |
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