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Theorem nmdvr 24635

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 24628	. . . . . 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 20372	. . . . 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 24629	. . . 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 24634	. . . . 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 7383	. . 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 20383	. . . 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 6845	. 2 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘(𝐴 / 𝐵)) = (𝑁‘(𝐴(._r‘𝑅)((inv_r‘𝑅)‘𝐵))))
24		nrgngp 24627	. . . . . 6 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
25	24	ad2antrr 727	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ NrmGrp)
26	8, 11	nmcl 24581	. . . . 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 11173	. . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐴) ∈ ℂ)
29	8, 6	unitss 20356	. . . . . 6 ⊢ 𝑈 ⊆ 𝑋
30	29, 5	sselid 3920	. . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝐵 ∈ 𝑋)
31	8, 11	nmcl 24581	. . . . 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 11173	. . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐵) ∈ ℂ)
34	11, 6	unitnmn0 24633	. . . . 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 11934	. 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 6499 (class class class)co 7367 ℝcr 11037 0cc0 11038 1c1 11039 · cmul 11043 / cdiv 11807 Basecbs 17179 ._rcmulr 17221 Ringcrg 20214 Unitcui 20335 inv_rcinvr 20367 /_rcdvr 20380 NzRingcnzr 20489 normcnm 24541 NrmGrpcngp 24542 NrmRingcnrg 24544

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

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 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ico 13304 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-0g 17404 df-topgen 17406 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-grp 18912 df-minusg 18913 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-nzr 20490 df-abv 20786 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-xms 24285 df-ms 24286 df-nm 24547 df-ngp 24548 df-nrg 24550

This theorem is referenced by: qqhnm 34134

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