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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 2734 | . . . 4 โข (๐ โ ๐ฅ = ๐ฅ) | |
4 | eqidd 2734 | . . . . 5 โข (๐ โ (Baseโ๐พ) = (Baseโ๐พ)) | |
5 | nmpropd.2 | . . . . . 6 โข (๐ โ (+gโ๐พ) = (+gโ๐ฟ)) | |
6 | 5 | oveqdr 7389 | . . . . 5 โข ((๐ โง (๐ฅ โ (Baseโ๐พ) โง ๐ฆ โ (Baseโ๐พ))) โ (๐ฅ(+gโ๐พ)๐ฆ) = (๐ฅ(+gโ๐ฟ)๐ฆ)) |
7 | 4, 1, 6 | grpidpropd 18525 | . . . 4 โข (๐ โ (0gโ๐พ) = (0gโ๐ฟ)) |
8 | 2, 3, 7 | oveq123d 7382 | . . 3 โข (๐ โ (๐ฅ(distโ๐พ)(0gโ๐พ)) = (๐ฅ(distโ๐ฟ)(0gโ๐ฟ))) |
9 | 1, 8 | mpteq12dv 5200 | . 2 โข (๐ โ (๐ฅ โ (Baseโ๐พ) โฆ (๐ฅ(distโ๐พ)(0gโ๐พ))) = (๐ฅ โ (Baseโ๐ฟ) โฆ (๐ฅ(distโ๐ฟ)(0gโ๐ฟ)))) |
10 | eqid 2733 | . . 3 โข (normโ๐พ) = (normโ๐พ) | |
11 | eqid 2733 | . . 3 โข (Baseโ๐พ) = (Baseโ๐พ) | |
12 | eqid 2733 | . . 3 โข (0gโ๐พ) = (0gโ๐พ) | |
13 | eqid 2733 | . . 3 โข (distโ๐พ) = (distโ๐พ) | |
14 | 10, 11, 12, 13 | nmfval 23967 | . 2 โข (normโ๐พ) = (๐ฅ โ (Baseโ๐พ) โฆ (๐ฅ(distโ๐พ)(0gโ๐พ))) |
15 | eqid 2733 | . . 3 โข (normโ๐ฟ) = (normโ๐ฟ) | |
16 | eqid 2733 | . . 3 โข (Baseโ๐ฟ) = (Baseโ๐ฟ) | |
17 | eqid 2733 | . . 3 โข (0gโ๐ฟ) = (0gโ๐ฟ) | |
18 | eqid 2733 | . . 3 โข (distโ๐ฟ) = (distโ๐ฟ) | |
19 | 15, 16, 17, 18 | nmfval 23967 | . 2 โข (normโ๐ฟ) = (๐ฅ โ (Baseโ๐ฟ) โฆ (๐ฅ(distโ๐ฟ)(0gโ๐ฟ))) |
20 | 9, 14, 19 | 3eqtr4g 2798 | 1 โข (๐ โ (normโ๐พ) = (normโ๐ฟ)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 397 = wceq 1542 โ wcel 2107 โฆ cmpt 5192 โcfv 6500 (class class class)co 7361 Basecbs 17091 +gcplusg 17141 distcds 17150 0gc0g 17329 normcnm 23955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 df-0g 17331 df-nm 23961 |
This theorem is referenced by: sranlm 24071 rlmnm 24076 zlmnm 32611 |
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