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Theorem tcphcphlem2 25289

Description: Lemma for tcphcph 25290: homogeneity. (Contributed by Mario Carneiro, 8-Oct-2015.)

**Hypotheses**
Ref	Expression
tcphval.n	⊢ 𝐺 = (toℂPreHil‘𝑊)
tcphcph.v	⊢ 𝑉 = (Base‘𝑊)
tcphcph.f	⊢ 𝐹 = (Scalar‘𝑊)
tcphcph.1	⊢ (𝜑 → 𝑊 ∈ PreHil)
tcphcph.2	⊢ (𝜑 → 𝐹 = (ℂ_fld ↾_s 𝐾))
tcphcph.h	⊢ , = (·_𝑖‘𝑊)
tcphcph.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾)
tcphcph.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
tcphcph.k	⊢ 𝐾 = (Base‘𝐹)
tcphcph.s	⊢ · = ( ·_𝑠 ‘𝑊)
tcphcphlem2.3	⊢ (𝜑 → 𝑋 ∈ 𝐾)
tcphcphlem2.4	⊢ (𝜑 → 𝑌 ∈ 𝑉)

**Assertion**
Ref	Expression
tcphcphlem2	⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = ((abs‘𝑋) · (√‘(𝑌 , 𝑌))))

Distinct variable groups: 𝑥, , 𝑥,𝐹 𝑥,𝐺 𝑥,𝑉 𝜑,𝑥 𝑥,𝑊 𝑥, · 𝑥,𝑋 𝑥,𝑌 Allowed substitution hint: 𝐾(𝑥)

**Proof of Theorem tcphcphlem2**
Step	Hyp	Ref	Expression
1		tcphval.n	. . . . . . 7 ⊢ 𝐺 = (toℂPreHil‘𝑊)
2		tcphcph.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
3		tcphcph.f	. . . . . . 7 ⊢ 𝐹 = (Scalar‘𝑊)
4		tcphcph.1	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ PreHil)
5		tcphcph.2	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (ℂ_fld ↾_s 𝐾))
6	1, 2, 3, 4, 5	phclm 25285	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂMod)
7		tcphcph.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐹)
8	3, 7	clmsscn 25131	. . . . . 6 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ)
9	6, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ⊆ ℂ)
10		tcphcphlem2.3	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐾)
11	9, 10	sseldd 4009	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ)
12	11	cjmulrcld 15255	. . 3 ⊢ (𝜑 → (𝑋 · (∗‘𝑋)) ∈ ℝ)
13	11	cjmulge0d 15257	. . 3 ⊢ (𝜑 → 0 ≤ (𝑋 · (∗‘𝑋)))
14		tcphcphlem2.4	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15		tcphcph.h	. . . . 5 ⊢ , = (·_𝑖‘𝑊)
16	1, 2, 3, 4, 5, 15	tcphcphlem3 25286	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ)
17	14, 16	mpdan 686	. . 3 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ)
18		oveq12 7457	. . . . . 6 ⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌))
19	18	anidms 566	. . . . 5 ⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌))
20	19	breq2d 5178	. . . 4 ⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌)))
21		tcphcph.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
22	21	ralrimiva 3152	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥))
23	20, 22, 14	rspcdva 3636	. . 3 ⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌))
24	12, 13, 17, 23	sqrtmuld 15473	. 2 ⊢ (𝜑 → (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌))))
25		phllmod 21671	. . . . . . 7 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
26	4, 25	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod)
27		tcphcph.s	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝑊)
28	2, 3, 27, 7	lmodvscl 20898	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) ∈ 𝑉)
29	26, 10, 14, 28	syl3anc 1371	. . . . 5 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑉)
30		eqid 2740	. . . . . 6 ⊢ (._r‘𝐹) = (._r‘𝐹)
31		eqid 2740	. . . . . 6 ⊢ (_𝑟‘𝐹) = (_𝑟‘𝐹)
32	3, 15, 2, 7, 27, 30, 31	ipassr 21687	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ ((𝑋 · 𝑌) ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾)) → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = (((𝑋 · 𝑌) , 𝑌)(._r‘𝐹)((*_𝑟‘𝐹)‘𝑋)))
33	4, 29, 14, 10, 32	syl13anc 1372	. . . 4 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = (((𝑋 · 𝑌) , 𝑌)(._r‘𝐹)((*_𝑟‘𝐹)‘𝑋)))
34	3	clmmul 25127	. . . . . 6 ⊢ (𝑊 ∈ ℂMod → · = (._r‘𝐹))
35	6, 34	syl 17	. . . . 5 ⊢ (𝜑 → · = (._r‘𝐹))
36	35	oveqd 7465	. . . . . 6 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = (𝑋(._r‘𝐹)(𝑌 , 𝑌)))
37	3, 15, 2, 7, 27, 30	ipass 21686	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑌) , 𝑌) = (𝑋(._r‘𝐹)(𝑌 , 𝑌)))
38	4, 10, 14, 14, 37	syl13anc 1372	. . . . . 6 ⊢ (𝜑 → ((𝑋 · 𝑌) , 𝑌) = (𝑋(._r‘𝐹)(𝑌 , 𝑌)))
39	36, 38	eqtr4d 2783	. . . . 5 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = ((𝑋 · 𝑌) , 𝑌))
40	3	clmcj 25128	. . . . . . 7 ⊢ (𝑊 ∈ ℂMod → ∗ = (*_𝑟‘𝐹))
41	6, 40	syl 17	. . . . . 6 ⊢ (𝜑 → ∗ = (*_𝑟‘𝐹))
42	41	fveq1d 6922	. . . . 5 ⊢ (𝜑 → (∗‘𝑋) = ((*_𝑟‘𝐹)‘𝑋))
43	35, 39, 42	oveq123d 7469	. . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = (((𝑋 · 𝑌) , 𝑌)(._r‘𝐹)((*_𝑟‘𝐹)‘𝑋)))
44	17	recnd 11318	. . . . 5 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ)
45	11	cjcld 15245	. . . . 5 ⊢ (𝜑 → (∗‘𝑋) ∈ ℂ)
46	11, 44, 45	mul32d 11500	. . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌)))
47	33, 43, 46	3eqtr2d 2786	. . 3 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌)))
48	47	fveq2d 6924	. 2 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))))
49		absval 15287	. . . 4 ⊢ (𝑋 ∈ ℂ → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋))))
50	11, 49	syl 17	. . 3 ⊢ (𝜑 → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋))))
51	50	oveq1d 7463	. 2 ⊢ (𝜑 → ((abs‘𝑋) · (√‘(𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌))))
52	24, 48, 51	3eqtr4d 2790	1 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = ((abs‘𝑋) · (√‘(𝑌 , 𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ⊆ wss 3976 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 · cmul 11189 ≤ cle 11325 ∗ccj 15145 √csqrt 15282 abscabs 15283 Basecbs 17258 ↾_s cress 17287 ._rcmulr 17312 *_𝑟cstv 17313 Scalarcsca 17314 ·_𝑠 cvsca 17315 ·_𝑖cip 17316 LModclmod 20880 ℂ_fldccnfld 21387 PreHilcphl 21665 ℂModcclm 25114 toℂPreHilctcph 25220

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-fz 13568 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-0g 17501 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-grp 18976 df-minusg 18977 df-subg 19163 df-ghm 19253 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-rhm 20498 df-subrg 20597 df-drng 20753 df-staf 20862 df-srng 20863 df-lmod 20882 df-lmhm 21044 df-lvec 21125 df-sra 21195 df-rgmod 21196 df-cnfld 21388 df-phl 21667 df-clm 25115

This theorem is referenced by: tcphcph 25290

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