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Theorem nmdvr 24578

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 766	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ NrmRing)
2		simprl 770	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝐴 ∈ 𝑋)
3		nrgring 24571	. . . . . 6 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring)
4	3	ad2antrr 726	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ Ring)
5		simprr 772	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝐵 ∈ 𝑈)
6		nmdvr.u	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑅)
7		eqid 2730	. . . . . 6 ⊢ (inv_r‘𝑅) = (inv_r‘𝑅)
8		nmdvr.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑅)
9	6, 7, 8	ringinvcl 20303	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐵 ∈ 𝑈) → ((inv_r‘𝑅)‘𝐵) ∈ 𝑋)
10	4, 5, 9	syl2anc 584	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → ((inv_r‘𝑅)‘𝐵) ∈ 𝑋)
11		nmdvr.n	. . . . 5 ⊢ 𝑁 = (norm‘𝑅)
12		eqid 2730	. . . . 5 ⊢ (._r‘𝑅) = (._r‘𝑅)
13	8, 11, 12	nmmul 24572	. . . 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 1373	. . 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 768	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ NzRing)
16	11, 6, 7	nminvr 24577	. . . . 5 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐵 ∈ 𝑈) → (𝑁‘((inv_r‘𝑅)‘𝐵)) = (1 / (𝑁‘𝐵)))
17	1, 15, 5, 16	syl3anc 1373	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘((inv_r‘𝑅)‘𝐵)) = (1 / (𝑁‘𝐵)))
18	17	oveq2d 7357	. . 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 2765	. 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 20314	. . . 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 6821	. 2 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘(𝐴 / 𝐵)) = (𝑁‘(𝐴(._r‘𝑅)((inv_r‘𝑅)‘𝐵))))
24		nrgngp 24570	. . . . . 6 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
25	24	ad2antrr 726	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ NrmGrp)
26	8, 11	nmcl 24524	. . . . 5 ⊢ ((𝑅 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ)
27	25, 2, 26	syl2anc 584	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐴) ∈ ℝ)
28	27	recnd 11132	. . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐴) ∈ ℂ)
29	8, 6	unitss 20287	. . . . . 6 ⊢ 𝑈 ⊆ 𝑋
30	29, 5	sselid 3930	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝐵 ∈ 𝑋)
31	8, 11	nmcl 24524	. . . . 5 ⊢ ((𝑅 ∈ NrmGrp ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ ℝ)
32	25, 30, 31	syl2anc 584	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐵) ∈ ℝ)
33	32	recnd 11132	. . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐵) ∈ ℂ)
34	11, 6	unitnmn0 24576	. . . . 5 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐵 ∈ 𝑈) → (𝑁‘𝐵) ≠ 0)
35	34	3expa 1118	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝐵 ∈ 𝑈) → (𝑁‘𝐵) ≠ 0)
36	35	adantrl 716	. . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐵) ≠ 0)
37	28, 33, 36	divrecd 11892	. 2 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → ((𝑁‘𝐴) / (𝑁‘𝐵)) = ((𝑁‘𝐴) · (1 / (𝑁‘𝐵))))
38	19, 23, 37	3eqtr4d 2775	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 1541 ∈ wcel 2110 ≠ wne 2926 ‘cfv 6477 (class class class)co 7341 ℝcr 10997 0cc0 10998 1c1 10999 · cmul 11003 / cdiv 11766 Basecbs 17112 ._rcmulr 17154 Ringcrg 20144 Unitcui 20266 inv_rcinvr 20298 /_rcdvr 20311 NzRingcnzr 20420 normcnm 24484 NrmGrpcngp 24485 NrmRingcnrg 24487

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-er 8617 df-map 8747 df-en 8865 df-dom 8866 df-sdom 8867 df-sup 9321 df-inf 9322 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-n0 12374 df-z 12461 df-uz 12725 df-q 12839 df-rp 12883 df-xneg 13003 df-xadd 13004 df-xmul 13005 df-ico 13243 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-0g 17337 df-topgen 17339 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-grp 18841 df-minusg 18842 df-cmn 19687 df-abl 19688 df-mgp 20052 df-rng 20064 df-ur 20093 df-ring 20146 df-oppr 20248 df-dvdsr 20268 df-unit 20269 df-invr 20299 df-dvr 20312 df-nzr 20421 df-abv 20717 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-top 22802 df-topon 22819 df-topsp 22841 df-bases 22854 df-xms 24228 df-ms 24229 df-nm 24490 df-ngp 24491 df-nrg 24493

This theorem is referenced by: qqhnm 33993

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