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Mirrors > Home > MPE Home > Th. List > gsumf1o | Structured version Visualization version Unicode version |
Description: Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.) |
Ref | Expression |
---|---|
gsumcl.b |
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gsumcl.z |
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gsumcl.g |
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gsumcl.a |
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gsumcl.f |
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gsumcl.w |
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gsumf1o.h |
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Ref | Expression |
---|---|
gsumf1o |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumcl.b |
. 2
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2 | gsumcl.z |
. 2
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3 | eqid 2451 |
. 2
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4 | gsumcl.g |
. . 3
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5 | cmnmnd 17445 |
. . 3
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6 | 4, 5 | syl 17 |
. 2
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7 | gsumcl.a |
. 2
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8 | gsumcl.f |
. 2
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9 | 1, 3, 4, 8 | cntzcmnf 17483 |
. 2
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10 | gsumcl.w |
. 2
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11 | gsumf1o.h |
. 2
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12 | 1, 2, 3, 6, 7, 8, 9, 10, 11 | gsumzf1o 17546 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-cnex 9595 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 ax-pre-mulgt0 9616 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-int 4235 df-iun 4280 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-se 4794 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-isom 5591 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-om 6693 df-1st 6793 df-2nd 6794 df-supp 6915 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-1o 7182 df-oadd 7186 df-er 7363 df-en 7570 df-dom 7571 df-sdom 7572 df-fin 7573 df-fsupp 7884 df-oi 8025 df-card 8373 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-sub 9862 df-neg 9863 df-nn 10610 df-n0 10870 df-z 10938 df-uz 11160 df-fz 11785 df-fzo 11916 df-seq 12214 df-hash 12516 df-0g 15340 df-gsum 15341 df-mgm 16488 df-sgrp 16527 df-mnd 16537 df-cntz 16971 df-cmn 17432 |
This theorem is referenced by: gsummptshft 17569 gsummptf1o 17595 gsummptfif1o 17600 gsum2dlem2 17603 gsumcom2 17607 psrass1lem 18601 psrcom 18633 psropprmul 18831 coe1mul2 18862 ply1coe 18889 ply1coeOLD 18890 tsmsf1o 21159 lgseisenlem3 24279 gsummpt2d 28544 |
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