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Mirrors > Home > MPE Home > Th. List > add32d | Structured version 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 9693 |
. 2
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5 | 1, 2, 3, 4 | syl3anc 1219 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4520 ax-nul 4528 ax-pow 4577 ax-pr 4638 ax-un 6481 ax-resscn 9449 ax-1cn 9450 ax-icn 9451 ax-addcl 9452 ax-addrcl 9453 ax-mulcl 9454 ax-mulrcl 9455 ax-mulcom 9456 ax-addass 9457 ax-mulass 9458 ax-distr 9459 ax-i2m1 9460 ax-1ne0 9461 ax-1rid 9462 ax-rnegex 9463 ax-rrecex 9464 ax-cnre 9465 ax-pre-lttri 9466 ax-pre-lttrn 9467 ax-pre-ltadd 9468 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2649 df-nel 2650 df-ral 2803 df-rex 2804 df-rab 2807 df-v 3078 df-sbc 3293 df-csb 3395 df-dif 3438 df-un 3440 df-in 3442 df-ss 3449 df-nul 3745 df-if 3899 df-pw 3969 df-sn 3985 df-pr 3987 df-op 3991 df-uni 4199 df-br 4400 df-opab 4458 df-mpt 4459 df-id 4743 df-po 4748 df-so 4749 df-xp 4953 df-rel 4954 df-cnv 4955 df-co 4956 df-dm 4957 df-rn 4958 df-res 4959 df-ima 4960 df-iota 5488 df-fun 5527 df-fn 5528 df-f 5529 df-f1 5530 df-fo 5531 df-f1o 5532 df-fv 5533 df-ov 6202 df-er 7210 df-en 7420 df-dom 7421 df-sdom 7422 df-pnf 9530 df-mnf 9531 df-ltxr 9533 |
This theorem is referenced by: nppcan 9741 muladd 9887 fladdz 11786 zesq 12103 abstri 12935 iseraltlem3 13278 sadadd2lem 13772 pythagtriplem1 14000 pythagtriplem12 14010 vdwlem2 14160 vdwlem6 14164 vdwlem8 14166 tchcphlem1 20881 uniioombllem5 21199 heron 22365 dcubic1 22372 mulog2sumlem1 22915 chpdifbndlem1 22934 selberg34r 22952 pntlemr 22983 brbtwn2 23302 axpasch 23338 lgamcvg2 27184 subfacval2 27218 cnapbmcpd 30316 clwwisshclwwlem1 30616 |
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