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Theorem nmdvr 22384

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 789	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ NrmRing)
2		simprl 793	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝐴 ∈ 𝑋)
3		nrgring 22377	. . . . . 6 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring)
4	3	ad2antrr 761	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ Ring)
5		simprr 795	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝐵 ∈ 𝑈)
6		nmdvr.u	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑅)
7		eqid 2621	. . . . . 6 ⊢ (inv_r‘𝑅) = (inv_r‘𝑅)
8		nmdvr.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑅)
9	6, 7, 8	ringinvcl 18597	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐵 ∈ 𝑈) → ((inv_r‘𝑅)‘𝐵) ∈ 𝑋)
10	4, 5, 9	syl2anc 692	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → ((inv_r‘𝑅)‘𝐵) ∈ 𝑋)
11		nmdvr.n	. . . . 5 ⊢ 𝑁 = (norm‘𝑅)
12		eqid 2621	. . . . 5 ⊢ (._r‘𝑅) = (._r‘𝑅)
13	8, 11, 12	nmmul 22378	. . . 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 1323	. . 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 791	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ NzRing)
16	11, 6, 7	nminvr 22383	. . . . 5 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐵 ∈ 𝑈) → (𝑁‘((inv_r‘𝑅)‘𝐵)) = (1 / (𝑁‘𝐵)))
17	1, 15, 5, 16	syl3anc 1323	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘((inv_r‘𝑅)‘𝐵)) = (1 / (𝑁‘𝐵)))
18	17	oveq2d 6620	. . 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 2655	. 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 18606	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈) → (𝐴 / 𝐵) = (𝐴(._r‘𝑅)((inv_r‘𝑅)‘𝐵)))
22	21	adantl 482	. . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝐴 / 𝐵) = (𝐴(._r‘𝑅)((inv_r‘𝑅)‘𝐵)))
23	22	fveq2d 6152	. 2 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘(𝐴 / 𝐵)) = (𝑁‘(𝐴(._r‘𝑅)((inv_r‘𝑅)‘𝐵))))
24		nrgngp 22376	. . . . . 6 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
25	24	ad2antrr 761	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ NrmGrp)
26	8, 11	nmcl 22330	. . . . 5 ⊢ ((𝑅 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ)
27	25, 2, 26	syl2anc 692	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐴) ∈ ℝ)
28	27	recnd 10012	. . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐴) ∈ ℂ)
29	8, 6	unitss 18581	. . . . . 6 ⊢ 𝑈 ⊆ 𝑋
30	29, 5	sseldi 3581	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝐵 ∈ 𝑋)
31	8, 11	nmcl 22330	. . . . 5 ⊢ ((𝑅 ∈ NrmGrp ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ ℝ)
32	25, 30, 31	syl2anc 692	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐵) ∈ ℝ)
33	32	recnd 10012	. . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐵) ∈ ℂ)
34	11, 6	unitnmn0 22382	. . . . 5 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐵 ∈ 𝑈) → (𝑁‘𝐵) ≠ 0)
35	34	3expa 1262	. . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝐵 ∈ 𝑈) → (𝑁‘𝐵) ≠ 0)
36	35	adantrl 751	. . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐵) ≠ 0)
37	28, 33, 36	divrecd 10748	. 2 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → ((𝑁‘𝐴) / (𝑁‘𝐵)) = ((𝑁‘𝐴) · (1 / (𝑁‘𝐵))))
38	19, 23, 37	3eqtr4d 2665	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 384 = wceq 1480 ∈ wcel 1987 ≠ wne 2790 ‘cfv 5847 (class class class)co 6604 ℝcr 9879 0cc0 9880 1c1 9881 · cmul 9885 / cdiv 10628 Basecbs 15781 ._rcmulr 15863 Ringcrg 18468 Unitcui 18560 inv_rcinvr 18592 /_rcdvr 18603 NzRingcnzr 19176 normcnm 22291 NrmGrpcngp 22292 NrmRingcnrg 22294

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1719 ax-4 1734 ax-5 1836 ax-6 1885 ax-7 1932 ax-8 1989 ax-9 1996 ax-10 2016 ax-11 2031 ax-12 2044 ax-13 2245 ax-ext 2601 ax-rep 4731 ax-sep 4741 ax-nul 4749 ax-pow 4803 ax-pr 4867 ax-un 6902 ax-cnex 9936 ax-resscn 9937 ax-1cn 9938 ax-icn 9939 ax-addcl 9940 ax-addrcl 9941 ax-mulcl 9942 ax-mulrcl 9943 ax-mulcom 9944 ax-addass 9945 ax-mulass 9946 ax-distr 9947 ax-i2m1 9948 ax-1ne0 9949 ax-1rid 9950 ax-rnegex 9951 ax-rrecex 9952 ax-cnre 9953 ax-pre-lttri 9954 ax-pre-lttrn 9955 ax-pre-ltadd 9956 ax-pre-mulgt0 9957 ax-pre-sup 9958

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1037 df-3an 1038 df-tru 1483 df-ex 1702 df-nf 1707 df-sb 1878 df-eu 2473 df-mo 2474 df-clab 2608 df-cleq 2614 df-clel 2617 df-nfc 2750 df-ne 2791 df-nel 2894 df-ral 2912 df-rex 2913 df-reu 2914 df-rmo 2915 df-rab 2916 df-v 3188 df-sbc 3418 df-csb 3515 df-dif 3558 df-un 3560 df-in 3562 df-ss 3569 df-pss 3571 df-nul 3892 df-if 4059 df-pw 4132 df-sn 4149 df-pr 4151 df-tp 4153 df-op 4155 df-uni 4403 df-iun 4487 df-br 4614 df-opab 4674 df-mpt 4675 df-tr 4713 df-eprel 4985 df-id 4989 df-po 4995 df-so 4996 df-fr 5033 df-we 5035 df-xp 5080 df-rel 5081 df-cnv 5082 df-co 5083 df-dm 5084 df-rn 5085 df-res 5086 df-ima 5087 df-pred 5639 df-ord 5685 df-on 5686 df-lim 5687 df-suc 5688 df-iota 5810 df-fun 5849 df-fn 5850 df-f 5851 df-f1 5852 df-fo 5853 df-f1o 5854 df-fv 5855 df-riota 6565 df-ov 6607 df-oprab 6608 df-mpt2 6609 df-om 7013 df-1st 7113 df-2nd 7114 df-tpos 7297 df-wrecs 7352 df-recs 7413 df-rdg 7451 df-er 7687 df-map 7804 df-en 7900 df-dom 7901 df-sdom 7902 df-sup 8292 df-inf 8293 df-pnf 10020 df-mnf 10021 df-xr 10022 df-ltxr 10023 df-le 10024 df-sub 10212 df-neg 10213 df-div 10629 df-nn 10965 df-2 11023 df-3 11024 df-n0 11237 df-z 11322 df-uz 11632 df-q 11733 df-rp 11777 df-xneg 11890 df-xadd 11891 df-xmul 11892 df-ico 12123 df-ndx 15784 df-slot 15785 df-base 15786 df-sets 15787 df-ress 15788 df-plusg 15875 df-mulr 15876 df-0g 16023 df-topgen 16025 df-mgm 17163 df-sgrp 17205 df-mnd 17216 df-grp 17346 df-minusg 17347 df-mgp 18411 df-ur 18423 df-ring 18470 df-oppr 18544 df-dvdsr 18562 df-unit 18563 df-invr 18593 df-dvr 18604 df-abv 18738 df-nzr 19177 df-psmet 19657 df-xmet 19658 df-met 19659 df-bl 19660 df-mopn 19661 df-top 20621 df-bases 20622 df-topon 20623 df-topsp 20624 df-xms 22035 df-ms 22036 df-nm 22297 df-ngp 22298 df-nrg 22300

This theorem is referenced by: qqhnm 29816

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