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Theorem nmdvr 24619

Description: The norm of a division in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)

**Hypotheses**
Ref	Expression
nmdvr.x	⊢ 𝑋 = (Base‘𝑅)
nmdvr.n	⊢ 𝑁 = (norm‘𝑅)
nmdvr.u	⊢ 𝑈 = (Unit‘𝑅)
nmdvr.d	⊢ / = (/_r‘𝑅)

**Assertion**
Ref	Expression
nmdvr	⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘(𝐴 / 𝐵)) = ((𝑁‘𝐴) / (𝑁‘𝐵)))

**Proof of Theorem nmdvr**
Step	Hyp	Ref	Expression
1		simpll 767	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ NrmRing)
2		simprl 771	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝐴 ∈ 𝑋)
3		nrgring 24612	. . . . . 6 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring)
4	3	ad2antrr 727	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ Ring)
5		simprr 773	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝐵 ∈ 𝑈)
6		nmdvr.u	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑅)
7		eqid 2737	. . . . . 6 ⊢ (inv_r‘𝑅) = (inv_r‘𝑅)
8		nmdvr.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑅)
9	6, 7, 8	ringinvcl 20333	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐵 ∈ 𝑈) → ((inv_r‘𝑅)‘𝐵) ∈ 𝑋)
10	4, 5, 9	syl2anc 585	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → ((inv_r‘𝑅)‘𝐵) ∈ 𝑋)
11		nmdvr.n	. . . . 5 ⊢ 𝑁 = (norm‘𝑅)
12		eqid 2737	. . . . 5 ⊢ (._r‘𝑅) = (._r‘𝑅)
13	8, 11, 12	nmmul 24613	. . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝐴 ∈ 𝑋 ∧ ((inv_r‘𝑅)‘𝐵) ∈ 𝑋) → (𝑁‘(𝐴(._r‘𝑅)((inv_r‘𝑅)‘𝐵))) = ((𝑁‘𝐴) · (𝑁‘((inv_r‘𝑅)‘𝐵))))
14	1, 2, 10, 13	syl3anc 1374	. . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘(𝐴(._r‘𝑅)((inv_r‘𝑅)‘𝐵))) = ((𝑁‘𝐴) · (𝑁‘((inv_r‘𝑅)‘𝐵))))
15		simplr 769	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ NzRing)
16	11, 6, 7	nminvr 24618	. . . . 5 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐵 ∈ 𝑈) → (𝑁‘((inv_r‘𝑅)‘𝐵)) = (1 / (𝑁‘𝐵)))
17	1, 15, 5, 16	syl3anc 1374	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘((inv_r‘𝑅)‘𝐵)) = (1 / (𝑁‘𝐵)))
18	17	oveq2d 7377	. . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → ((𝑁‘𝐴) · (𝑁‘((inv_r‘𝑅)‘𝐵))) = ((𝑁‘𝐴) · (1 / (𝑁‘𝐵))))
19	14, 18	eqtrd 2772	. 2 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘(𝐴(._r‘𝑅)((inv_r‘𝑅)‘𝐵))) = ((𝑁‘𝐴) · (1 / (𝑁‘𝐵))))
20		nmdvr.d	. . . . 5 ⊢ / = (/_r‘𝑅)
21	8, 12, 6, 7, 20	dvrval 20344	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈) → (𝐴 / 𝐵) = (𝐴(._r‘𝑅)((inv_r‘𝑅)‘𝐵)))
22	21	adantl 481	. . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝐴 / 𝐵) = (𝐴(._r‘𝑅)((inv_r‘𝑅)‘𝐵)))
23	22	fveq2d 6839	. 2 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘(𝐴 / 𝐵)) = (𝑁‘(𝐴(._r‘𝑅)((inv_r‘𝑅)‘𝐵))))
24		nrgngp 24611	. . . . . 6 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
25	24	ad2antrr 727	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ NrmGrp)
26	8, 11	nmcl 24565	. . . . 5 ⊢ ((𝑅 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ)
27	25, 2, 26	syl2anc 585	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐴) ∈ ℝ)
28	27	recnd 11165	. . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐴) ∈ ℂ)
29	8, 6	unitss 20317	. . . . . 6 ⊢ 𝑈 ⊆ 𝑋
30	29, 5	sselid 3932	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝐵 ∈ 𝑋)
31	8, 11	nmcl 24565	. . . . 5 ⊢ ((𝑅 ∈ NrmGrp ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ ℝ)
32	25, 30, 31	syl2anc 585	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐵) ∈ ℝ)
33	32	recnd 11165	. . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐵) ∈ ℂ)
34	11, 6	unitnmn0 24617	. . . . 5 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐵 ∈ 𝑈) → (𝑁‘𝐵) ≠ 0)
35	34	3expa 1119	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝐵 ∈ 𝑈) → (𝑁‘𝐵) ≠ 0)
36	35	adantrl 717	. . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐵) ≠ 0)
37	28, 33, 36	divrecd 11925	. 2 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → ((𝑁‘𝐴) / (𝑁‘𝐵)) = ((𝑁‘𝐴) · (1 / (𝑁‘𝐵))))
38	19, 23, 37	3eqtr4d 2782	1 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘(𝐴 / 𝐵)) = ((𝑁‘𝐴) / (𝑁‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ‘cfv 6493 (class class class)co 7361 ℝcr 11030 0cc0 11031 1c1 11032 · cmul 11036 / cdiv 11799 Basecbs 17141 ._rcmulr 17183 Ringcrg 20173 Unitcui 20296 inv_rcinvr 20328 /_rcdvr 20341 NzRingcnzr 20450 normcnm 24525 NrmGrpcngp 24526 NrmRingcnrg 24528

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7683 ax-cnex 11087 ax-resscn 11088 ax-1cn 11089 ax-icn 11090 ax-addcl 11091 ax-addrcl 11092 ax-mulcl 11093 ax-mulrcl 11094 ax-mulcom 11095 ax-addass 11096 ax-mulass 11097 ax-distr 11098 ax-i2m1 11099 ax-1ne0 11100 ax-1rid 11101 ax-rnegex 11102 ax-rrecex 11103 ax-cnre 11104 ax-pre-lttri 11105 ax-pre-lttrn 11106 ax-pre-ltadd 11107 ax-pre-mulgt0 11108 ax-pre-sup 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-tpos 8171 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-er 8638 df-map 8770 df-en 8889 df-dom 8890 df-sdom 8891 df-sup 9350 df-inf 9351 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12151 df-2 12213 df-3 12214 df-n0 12407 df-z 12494 df-uz 12757 df-q 12867 df-rp 12911 df-xneg 13031 df-xadd 13032 df-xmul 13033 df-ico 13272 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17142 df-ress 17163 df-plusg 17195 df-mulr 17196 df-0g 17366 df-topgen 17368 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-grp 18871 df-minusg 18872 df-cmn 19716 df-abl 19717 df-mgp 20081 df-rng 20093 df-ur 20122 df-ring 20175 df-oppr 20278 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-dvr 20342 df-nzr 20451 df-abv 20747 df-psmet 21306 df-xmet 21307 df-met 21308 df-bl 21309 df-mopn 21310 df-top 22843 df-topon 22860 df-topsp 22882 df-bases 22895 df-xms 24269 df-ms 24270 df-nm 24531 df-ngp 24532 df-nrg 24534

This theorem is referenced by: qqhnm 34160

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