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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 2729 | . . . 4 โข (๐ โ ๐ฅ = ๐ฅ) | |
4 | eqidd 2729 | . . . . 5 โข (๐ โ (Baseโ๐พ) = (Baseโ๐พ)) | |
5 | nmpropd.2 | . . . . . 6 โข (๐ โ (+gโ๐พ) = (+gโ๐ฟ)) | |
6 | 5 | oveqdr 7448 | . . . . 5 โข ((๐ โง (๐ฅ โ (Baseโ๐พ) โง ๐ฆ โ (Baseโ๐พ))) โ (๐ฅ(+gโ๐พ)๐ฆ) = (๐ฅ(+gโ๐ฟ)๐ฆ)) |
7 | 4, 1, 6 | grpidpropd 18621 | . . . 4 โข (๐ โ (0gโ๐พ) = (0gโ๐ฟ)) |
8 | 2, 3, 7 | oveq123d 7441 | . . 3 โข (๐ โ (๐ฅ(distโ๐พ)(0gโ๐พ)) = (๐ฅ(distโ๐ฟ)(0gโ๐ฟ))) |
9 | 1, 8 | mpteq12dv 5239 | . 2 โข (๐ โ (๐ฅ โ (Baseโ๐พ) โฆ (๐ฅ(distโ๐พ)(0gโ๐พ))) = (๐ฅ โ (Baseโ๐ฟ) โฆ (๐ฅ(distโ๐ฟ)(0gโ๐ฟ)))) |
10 | eqid 2728 | . . 3 โข (normโ๐พ) = (normโ๐พ) | |
11 | eqid 2728 | . . 3 โข (Baseโ๐พ) = (Baseโ๐พ) | |
12 | eqid 2728 | . . 3 โข (0gโ๐พ) = (0gโ๐พ) | |
13 | eqid 2728 | . . 3 โข (distโ๐พ) = (distโ๐พ) | |
14 | 10, 11, 12, 13 | nmfval 24496 | . 2 โข (normโ๐พ) = (๐ฅ โ (Baseโ๐พ) โฆ (๐ฅ(distโ๐พ)(0gโ๐พ))) |
15 | eqid 2728 | . . 3 โข (normโ๐ฟ) = (normโ๐ฟ) | |
16 | eqid 2728 | . . 3 โข (Baseโ๐ฟ) = (Baseโ๐ฟ) | |
17 | eqid 2728 | . . 3 โข (0gโ๐ฟ) = (0gโ๐ฟ) | |
18 | eqid 2728 | . . 3 โข (distโ๐ฟ) = (distโ๐ฟ) | |
19 | 15, 16, 17, 18 | nmfval 24496 | . 2 โข (normโ๐ฟ) = (๐ฅ โ (Baseโ๐ฟ) โฆ (๐ฅ(distโ๐ฟ)(0gโ๐ฟ))) |
20 | 9, 14, 19 | 3eqtr4g 2793 | 1 โข (๐ โ (normโ๐พ) = (normโ๐ฟ)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 395 = wceq 1534 โ wcel 2099 โฆ cmpt 5231 โcfv 6548 (class class class)co 7420 Basecbs 17179 +gcplusg 17232 distcds 17241 0gc0g 17420 normcnm 24484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-0g 17422 df-nm 24490 |
This theorem is referenced by: sranlm 24600 rlmnm 24605 zlmnm 33567 |
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