![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nmpropd | Structured version Visualization version GIF version |
Description: Weak property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nmpropd.1 | โข (๐ โ (Baseโ๐พ) = (Baseโ๐ฟ)) |
nmpropd.2 | โข (๐ โ (+gโ๐พ) = (+gโ๐ฟ)) |
nmpropd.3 | โข (๐ โ (distโ๐พ) = (distโ๐ฟ)) |
Ref | Expression |
---|---|
nmpropd | โข (๐ โ (normโ๐พ) = (normโ๐ฟ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmpropd.1 | . . 3 โข (๐ โ (Baseโ๐พ) = (Baseโ๐ฟ)) | |
2 | nmpropd.3 | . . . 4 โข (๐ โ (distโ๐พ) = (distโ๐ฟ)) | |
3 | eqidd 2725 | . . . 4 โข (๐ โ ๐ฅ = ๐ฅ) | |
4 | eqidd 2725 | . . . . 5 โข (๐ โ (Baseโ๐พ) = (Baseโ๐พ)) | |
5 | nmpropd.2 | . . . . . 6 โข (๐ โ (+gโ๐พ) = (+gโ๐ฟ)) | |
6 | 5 | oveqdr 7430 | . . . . 5 โข ((๐ โง (๐ฅ โ (Baseโ๐พ) โง ๐ฆ โ (Baseโ๐พ))) โ (๐ฅ(+gโ๐พ)๐ฆ) = (๐ฅ(+gโ๐ฟ)๐ฆ)) |
7 | 4, 1, 6 | grpidpropd 18591 | . . . 4 โข (๐ โ (0gโ๐พ) = (0gโ๐ฟ)) |
8 | 2, 3, 7 | oveq123d 7423 | . . 3 โข (๐ โ (๐ฅ(distโ๐พ)(0gโ๐พ)) = (๐ฅ(distโ๐ฟ)(0gโ๐ฟ))) |
9 | 1, 8 | mpteq12dv 5230 | . 2 โข (๐ โ (๐ฅ โ (Baseโ๐พ) โฆ (๐ฅ(distโ๐พ)(0gโ๐พ))) = (๐ฅ โ (Baseโ๐ฟ) โฆ (๐ฅ(distโ๐ฟ)(0gโ๐ฟ)))) |
10 | eqid 2724 | . . 3 โข (normโ๐พ) = (normโ๐พ) | |
11 | eqid 2724 | . . 3 โข (Baseโ๐พ) = (Baseโ๐พ) | |
12 | eqid 2724 | . . 3 โข (0gโ๐พ) = (0gโ๐พ) | |
13 | eqid 2724 | . . 3 โข (distโ๐พ) = (distโ๐พ) | |
14 | 10, 11, 12, 13 | nmfval 24441 | . 2 โข (normโ๐พ) = (๐ฅ โ (Baseโ๐พ) โฆ (๐ฅ(distโ๐พ)(0gโ๐พ))) |
15 | eqid 2724 | . . 3 โข (normโ๐ฟ) = (normโ๐ฟ) | |
16 | eqid 2724 | . . 3 โข (Baseโ๐ฟ) = (Baseโ๐ฟ) | |
17 | eqid 2724 | . . 3 โข (0gโ๐ฟ) = (0gโ๐ฟ) | |
18 | eqid 2724 | . . 3 โข (distโ๐ฟ) = (distโ๐ฟ) | |
19 | 15, 16, 17, 18 | nmfval 24441 | . 2 โข (normโ๐ฟ) = (๐ฅ โ (Baseโ๐ฟ) โฆ (๐ฅ(distโ๐ฟ)(0gโ๐ฟ))) |
20 | 9, 14, 19 | 3eqtr4g 2789 | 1 โข (๐ โ (normโ๐พ) = (normโ๐ฟ)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 395 = wceq 1533 โ wcel 2098 โฆ cmpt 5222 โcfv 6534 (class class class)co 7402 Basecbs 17149 +gcplusg 17202 distcds 17211 0gc0g 17390 normcnm 24429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-ov 7405 df-0g 17392 df-nm 24435 |
This theorem is referenced by: sranlm 24545 rlmnm 24550 zlmnm 33465 |
Copyright terms: Public domain | W3C validator |