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Theorem nmf2 23972
Description: The norm on a metric group is a function from the base set into the reals. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
nmf2.n 𝑁 = (normβ€˜π‘Š)
nmf2.x 𝑋 = (Baseβ€˜π‘Š)
nmf2.d 𝐷 = (distβ€˜π‘Š)
nmf2.e 𝐸 = (𝐷 β†Ύ (𝑋 Γ— 𝑋))
Assertion
Ref Expression
nmf2 ((π‘Š ∈ Grp ∧ 𝐸 ∈ (Metβ€˜π‘‹)) β†’ 𝑁:π‘‹βŸΆβ„)

Proof of Theorem nmf2
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 nmf2.n . . . 4 𝑁 = (normβ€˜π‘Š)
2 nmf2.x . . . 4 𝑋 = (Baseβ€˜π‘Š)
3 eqid 2733 . . . 4 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
4 nmf2.d . . . 4 𝐷 = (distβ€˜π‘Š)
5 nmf2.e . . . 4 𝐸 = (𝐷 β†Ύ (𝑋 Γ— 𝑋))
61, 2, 3, 4, 5nmfval2 23970 . . 3 (π‘Š ∈ Grp β†’ 𝑁 = (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐸(0gβ€˜π‘Š))))
76adantr 482 . 2 ((π‘Š ∈ Grp ∧ 𝐸 ∈ (Metβ€˜π‘‹)) β†’ 𝑁 = (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐸(0gβ€˜π‘Š))))
82, 3grpidcl 18786 . . . 4 (π‘Š ∈ Grp β†’ (0gβ€˜π‘Š) ∈ 𝑋)
9 metcl 23708 . . . . 5 ((𝐸 ∈ (Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋 ∧ (0gβ€˜π‘Š) ∈ 𝑋) β†’ (π‘₯𝐸(0gβ€˜π‘Š)) ∈ ℝ)
1093comr 1126 . . . 4 (((0gβ€˜π‘Š) ∈ 𝑋 ∧ 𝐸 ∈ (Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯𝐸(0gβ€˜π‘Š)) ∈ ℝ)
118, 10syl3an1 1164 . . 3 ((π‘Š ∈ Grp ∧ 𝐸 ∈ (Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯𝐸(0gβ€˜π‘Š)) ∈ ℝ)
12113expa 1119 . 2 (((π‘Š ∈ Grp ∧ 𝐸 ∈ (Metβ€˜π‘‹)) ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯𝐸(0gβ€˜π‘Š)) ∈ ℝ)
137, 12fmpt3d 7068 1 ((π‘Š ∈ Grp ∧ 𝐸 ∈ (Metβ€˜π‘‹)) β†’ 𝑁:π‘‹βŸΆβ„)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ↦ cmpt 5192   Γ— cxp 5635   β†Ύ cres 5639  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  β„cr 11058  Basecbs 17091  distcds 17150  0gc0g 17329  Grpcgrp 18756  Metcmet 20805  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  ax-cnex 11115  ax-resscn 11116
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-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  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-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773  df-0g 17331  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-grp 18759  df-met 20813  df-nm 23961
This theorem is referenced by:  isngp2  23976  isngp3  23977  nmf  23994
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