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Mirrors > Home > MPE Home > Th. List > gsumpropd | Structured version Visualization version Unicode version |
Description: The group sum depends only on the base set and additive operation. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 17316 etc. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Mario Carneiro, 18-Sep-2015.) |
Ref | Expression |
---|---|
gsumpropd.f | |
gsumpropd.g | |
gsumpropd.h | |
gsumpropd.b | |
gsumpropd.p |
Ref | Expression |
---|---|
gsumpropd | g g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumpropd.b | . . . . 5 | |
2 | gsumpropd.p | . . . . . . . . 9 | |
3 | 2 | oveqd 6667 | . . . . . . . 8 |
4 | 3 | eqeq1d 2624 | . . . . . . 7 |
5 | 2 | oveqd 6667 | . . . . . . . 8 |
6 | 5 | eqeq1d 2624 | . . . . . . 7 |
7 | 4, 6 | anbi12d 747 | . . . . . 6 |
8 | 1, 7 | raleqbidv 3152 | . . . . 5 |
9 | 1, 8 | rabeqbidv 3195 | . . . 4 |
10 | 9 | sseq2d 3633 | . . 3 |
11 | eqidd 2623 | . . . 4 | |
12 | 2 | oveqdr 6674 | . . . 4 |
13 | 11, 1, 12 | grpidpropd 17261 | . . 3 |
14 | 2 | seqeq2d 12808 | . . . . . . . . . 10 |
15 | 14 | fveq1d 6193 | . . . . . . . . 9 |
16 | 15 | eqeq2d 2632 | . . . . . . . 8 |
17 | 16 | anbi2d 740 | . . . . . . 7 |
18 | 17 | rexbidv 3052 | . . . . . 6 |
19 | 18 | exbidv 1850 | . . . . 5 |
20 | 19 | iotabidv 5872 | . . . 4 |
21 | 9 | difeq2d 3728 | . . . . . . . . . . . 12 |
22 | 21 | imaeq2d 5466 | . . . . . . . . . . 11 |
23 | 22 | fveq2d 6195 | . . . . . . . . . 10 |
24 | 23 | oveq2d 6666 | . . . . . . . . 9 |
25 | f1oeq2 6128 | . . . . . . . . 9 | |
26 | 24, 25 | syl 17 | . . . . . . . 8 |
27 | f1oeq3 6129 | . . . . . . . . 9 | |
28 | 22, 27 | syl 17 | . . . . . . . 8 |
29 | 26, 28 | bitrd 268 | . . . . . . 7 |
30 | 2 | seqeq2d 12808 | . . . . . . . . 9 |
31 | 30, 23 | fveq12d 6197 | . . . . . . . 8 |
32 | 31 | eqeq2d 2632 | . . . . . . 7 |
33 | 29, 32 | anbi12d 747 | . . . . . 6 |
34 | 33 | exbidv 1850 | . . . . 5 |
35 | 34 | iotabidv 5872 | . . . 4 |
36 | 20, 35 | ifeq12d 4106 | . . 3 |
37 | 10, 13, 36 | ifbieq12d 4113 | . 2 |
38 | eqid 2622 | . . 3 | |
39 | eqid 2622 | . . 3 | |
40 | eqid 2622 | . . 3 | |
41 | eqid 2622 | . . 3 | |
42 | eqidd 2623 | . . 3 | |
43 | gsumpropd.g | . . 3 | |
44 | gsumpropd.f | . . 3 | |
45 | eqidd 2623 | . . 3 | |
46 | 38, 39, 40, 41, 42, 43, 44, 45 | gsumvalx 17270 | . 2 g |
47 | eqid 2622 | . . 3 | |
48 | eqid 2622 | . . 3 | |
49 | eqid 2622 | . . 3 | |
50 | eqid 2622 | . . 3 | |
51 | eqidd 2623 | . . 3 | |
52 | gsumpropd.h | . . 3 | |
53 | 47, 48, 49, 50, 51, 52, 44, 45 | gsumvalx 17270 | . 2 g |
54 | 37, 46, 53 | 3eqtr4d 2666 | 1 g g |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 wral 2912 wrex 2913 crab 2916 cvv 3200 cdif 3571 wss 3574 cif 4086 ccnv 5113 cdm 5114 crn 5115 cima 5117 ccom 5118 cio 5849 wf1o 5887 cfv 5888 (class class class)co 6650 c1 9937 cuz 11687 cfz 12326 cseq 12801 chash 13117 cbs 15857 cplusg 15941 c0g 16100 g cgsu 16101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-seq 12802 df-0g 16102 df-gsum 16103 |
This theorem is referenced by: psropprmul 19608 ply1coe 19666 frlmgsum 20111 matgsum 20243 tsmspropd 21935 |
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