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Theorem tcphcphlem2 25376

Description: Lemma for tcphcph 25377: 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 25372	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂMod)
7		tcphcph.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐹)
8	3, 7	clmsscn 25219	. . . . . 6 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ)
9	6, 8	syl 18	. . . . 5 ⊢ (𝜑 → 𝐾 ⊆ ℂ)
10		tcphcphlem2.3	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐾)
11	9, 10	sseldd 3939	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ)
12	11	cjmulrcld 15259	. . 3 ⊢ (𝜑 → (𝑋 · (∗‘𝑋)) ∈ ℝ)
13	11	cjmulge0d 15261	. . 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 25373	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ)
17	14, 16	mpdan 699	. . 3 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ)
18		oveq12 7421	. . . . . 6 ⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌))
19	18	anidms 576	. . . . 5 ⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌))
20	19	breq2d 5122	. . . 4 ⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌)))
21		tcphcph.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
22	21	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥))
23	20, 22, 14	rspcdva 3583	. . 3 ⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌))
24	12, 13, 17, 23	sqrtmuld 15478	. 2 ⊢ (𝜑 → (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌))))
25		phllmod 21761	. . . . . . 7 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
26	4, 25	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod)
27		tcphcph.s	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝑊)
28	2, 3, 27, 7	lmodvscl 20980	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) ∈ 𝑉)
29	26, 10, 14, 28	syl3anc 1398	. . . . 5 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑉)
30		eqid 2763	. . . . . 6 ⊢ (._r‘𝐹) = (._r‘𝐹)
31		eqid 2763	. . . . . 6 ⊢ (_𝑟‘𝐹) = (_𝑟‘𝐹)
32	3, 15, 2, 7, 27, 30, 31	ipassr 21777	. . . . 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 1399	. . . 4 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = (((𝑋 · 𝑌) , 𝑌)(._r‘𝐹)((*_𝑟‘𝐹)‘𝑋)))
34	3	clmmul 25215	. . . . . 6 ⊢ (𝑊 ∈ ℂMod → · = (._r‘𝐹))
35	6, 34	syl 18	. . . . 5 ⊢ (𝜑 → · = (._r‘𝐹))
36	35	oveqd 7429	. . . . . 6 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = (𝑋(._r‘𝐹)(𝑌 , 𝑌)))
37	3, 15, 2, 7, 27, 30	ipass 21776	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑌) , 𝑌) = (𝑋(._r‘𝐹)(𝑌 , 𝑌)))
38	4, 10, 14, 14, 37	syl13anc 1399	. . . . . 6 ⊢ (𝜑 → ((𝑋 · 𝑌) , 𝑌) = (𝑋(._r‘𝐹)(𝑌 , 𝑌)))
39	36, 38	eqtr4d 2801	. . . . 5 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = ((𝑋 · 𝑌) , 𝑌))
40	3	clmcj 25216	. . . . . . 7 ⊢ (𝑊 ∈ ℂMod → ∗ = (*_𝑟‘𝐹))
41	6, 40	syl 18	. . . . . 6 ⊢ (𝜑 → ∗ = (*_𝑟‘𝐹))
42	41	fveq1d 6885	. . . . 5 ⊢ (𝜑 → (∗‘𝑋) = ((*_𝑟‘𝐹)‘𝑋))
43	35, 39, 42	oveq123d 7433	. . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = (((𝑋 · 𝑌) , 𝑌)(._r‘𝐹)((*_𝑟‘𝐹)‘𝑋)))
44	17	recnd 11238	. . . . 5 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ)
45	11	cjcld 15249	. . . . 5 ⊢ (𝜑 → (∗‘𝑋) ∈ ℂ)
46	11, 44, 45	mul32d 11421	. . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌)))
47	33, 43, 46	3eqtr2d 2804	. . 3 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌)))
48	47	fveq2d 6887	. 2 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))))
49		absval 15291	. . . 4 ⊢ (𝑋 ∈ ℂ → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋))))
50	11, 49	syl 18	. . 3 ⊢ (𝜑 → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋))))
51	50	oveq1d 7427	. 2 ⊢ (𝜑 → ((abs‘𝑋) · (√‘(𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌))))
52	24, 48, 51	3eqtr4d 2808	1 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = ((abs‘𝑋) · (√‘(𝑌 , 𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ⊆ wss 3906 class class class wbr 5110 ‘cfv 6538 (class class class)co 7412 ℂcc 11099 ℝcr 11100 0cc0 11101 · cmul 11106 ≤ cle 11245 ∗ccj 15149 √csqrt 15286 abscabs 15287 Basecbs 17270 ↾_s cress 17291 ._rcmulr 17312 *_𝑟cstv 17313 Scalarcsca 17314 ·_𝑠 cvsca 17315 ·_𝑖cip 17316 LModclmod 20962 ℂ_fldccnfld 21503 PreHilcphl 21755 ℂModcclm 25202 toℂPreHilctcph 25307

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179 ax-addf 11180 ax-mulf 11181

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-tpos 8223 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-map 8827 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-sup 9403 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-rp 13018 df-fz 13537 df-seq 14040 df-exp 14100 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 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 17495 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-mhm 18842 df-grp 19004 df-minusg 19005 df-subg 19190 df-ghm 19285 df-cmn 19853 df-abl 19854 df-mgp 20218 df-rng 20232 df-ur 20265 df-ring 20318 df-cring 20319 df-oppr 20420 df-dvdsr 20440 df-unit 20441 df-rhm 20555 df-subrg 20656 df-drng 20816 df-staf 20923 df-srng 20924 df-lmod 20964 df-lmhm 21124 df-lvec 21205 df-sra 21275 df-rgmod 21276 df-cnfld 21504 df-phl 21757 df-clm 25203

This theorem is referenced by: tcphcph 25377

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