Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > tsmsval2 | Structured version Visualization version Unicode version |
Description: Definition of the topological group sum(s) of a collection of values in the group with index set . (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tsmsval.b | |
tsmsval.j | |
tsmsval.s | |
tsmsval.l | |
tsmsval.g | |
tsmsval2.f | |
tsmsval2.a |
Ref | Expression |
---|---|
tsmsval2 | tsums g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tsms 21930 | . . 3 tsums g | |
2 | 1 | a1i 11 | . 2 tsums g |
3 | vex 3203 | . . . . . . 7 | |
4 | 3 | dmex 7099 | . . . . . 6 |
5 | 4 | pwex 4848 | . . . . 5 |
6 | 5 | inex1 4799 | . . . 4 |
7 | 6 | a1i 11 | . . 3 |
8 | simplrl 800 | . . . . . . 7 | |
9 | 8 | fveq2d 6195 | . . . . . 6 |
10 | tsmsval.j | . . . . . 6 | |
11 | 9, 10 | syl6eqr 2674 | . . . . 5 |
12 | id 22 | . . . . . . 7 | |
13 | simprr 796 | . . . . . . . . . . . 12 | |
14 | 13 | dmeqd 5326 | . . . . . . . . . . 11 |
15 | tsmsval2.a | . . . . . . . . . . . 12 | |
16 | 15 | adantr 481 | . . . . . . . . . . 11 |
17 | 14, 16 | eqtrd 2656 | . . . . . . . . . 10 |
18 | 17 | pweqd 4163 | . . . . . . . . 9 |
19 | 18 | ineq1d 3813 | . . . . . . . 8 |
20 | tsmsval.s | . . . . . . . 8 | |
21 | 19, 20 | syl6eqr 2674 | . . . . . . 7 |
22 | 12, 21 | sylan9eqr 2678 | . . . . . 6 |
23 | rabeq 3192 | . . . . . . . . . 10 | |
24 | 22, 23 | syl 17 | . . . . . . . . 9 |
25 | 22, 24 | mpteq12dv 4733 | . . . . . . . 8 |
26 | 25 | rneqd 5353 | . . . . . . 7 |
27 | tsmsval.l | . . . . . . 7 | |
28 | 26, 27 | syl6eqr 2674 | . . . . . 6 |
29 | 22, 28 | oveq12d 6668 | . . . . 5 |
30 | 11, 29 | oveq12d 6668 | . . . 4 |
31 | simplrr 801 | . . . . . . 7 | |
32 | 31 | reseq1d 5395 | . . . . . 6 |
33 | 8, 32 | oveq12d 6668 | . . . . 5 g g |
34 | 22, 33 | mpteq12dv 4733 | . . . 4 g g |
35 | 30, 34 | fveq12d 6197 | . . 3 g g |
36 | 7, 35 | csbied 3560 | . 2 g g |
37 | tsmsval.g | . . 3 | |
38 | elex 3212 | . . 3 | |
39 | 37, 38 | syl 17 | . 2 |
40 | tsmsval2.f | . . 3 | |
41 | elex 3212 | . . 3 | |
42 | 40, 41 | syl 17 | . 2 |
43 | fvexd 6203 | . 2 g | |
44 | 2, 36, 39, 42, 43 | ovmpt2d 6788 | 1 tsums g |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 crab 2916 cvv 3200 csb 3533 cin 3573 wss 3574 cpw 4158 cmpt 4729 cdm 5114 crn 5115 cres 5116 cfv 5888 (class class class)co 6650 cmpt2 6652 cfn 7955 cbs 15857 ctopn 16082 g cgsu 16101 cfg 19735 cflf 21739 tsums ctsu 21929 |
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-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-tsms 21930 |
This theorem is referenced by: tsmsval 21934 tsmspropd 21935 |
Copyright terms: Public domain | W3C validator |