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Theorem tcphcphlem2 23839

Description: Lemma for tcphcph 23840: 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 23835	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂMod)
7		tcphcph.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐹)
8	3, 7	clmsscn 23683	. . . . . 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 3968	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ)
12	11	cjmulrcld 14565	. . 3 ⊢ (𝜑 → (𝑋 · (∗‘𝑋)) ∈ ℝ)
13	11	cjmulge0d 14567	. . 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 23836	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ)
17	14, 16	mpdan 685	. . 3 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ)
18		oveq12 7165	. . . . . 6 ⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌))
19	18	anidms 569	. . . . 5 ⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌))
20	19	breq2d 5078	. . . 4 ⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌)))
21		tcphcph.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
22	21	ralrimiva 3182	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥))
23	20, 22, 14	rspcdva 3625	. . 3 ⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌))
24	12, 13, 17, 23	sqrtmuld 14784	. 2 ⊢ (𝜑 → (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌))))
25		phllmod 20774	. . . . . . 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 19651	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) ∈ 𝑉)
29	26, 10, 14, 28	syl3anc 1367	. . . . 5 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑉)
30		eqid 2821	. . . . . 6 ⊢ (._r‘𝐹) = (._r‘𝐹)
31		eqid 2821	. . . . . 6 ⊢ (_𝑟‘𝐹) = (_𝑟‘𝐹)
32	3, 15, 2, 7, 27, 30, 31	ipassr 20790	. . . . 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 1368	. . . 4 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = (((𝑋 · 𝑌) , 𝑌)(._r‘𝐹)((*_𝑟‘𝐹)‘𝑋)))
34	3	clmmul 23679	. . . . . 6 ⊢ (𝑊 ∈ ℂMod → · = (._r‘𝐹))
35	6, 34	syl 17	. . . . 5 ⊢ (𝜑 → · = (._r‘𝐹))
36	35	oveqd 7173	. . . . . 6 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = (𝑋(._r‘𝐹)(𝑌 , 𝑌)))
37	3, 15, 2, 7, 27, 30	ipass 20789	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑌) , 𝑌) = (𝑋(._r‘𝐹)(𝑌 , 𝑌)))
38	4, 10, 14, 14, 37	syl13anc 1368	. . . . . 6 ⊢ (𝜑 → ((𝑋 · 𝑌) , 𝑌) = (𝑋(._r‘𝐹)(𝑌 , 𝑌)))
39	36, 38	eqtr4d 2859	. . . . 5 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = ((𝑋 · 𝑌) , 𝑌))
40	3	clmcj 23680	. . . . . . 7 ⊢ (𝑊 ∈ ℂMod → ∗ = (*_𝑟‘𝐹))
41	6, 40	syl 17	. . . . . 6 ⊢ (𝜑 → ∗ = (*_𝑟‘𝐹))
42	41	fveq1d 6672	. . . . 5 ⊢ (𝜑 → (∗‘𝑋) = ((*_𝑟‘𝐹)‘𝑋))
43	35, 39, 42	oveq123d 7177	. . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = (((𝑋 · 𝑌) , 𝑌)(._r‘𝐹)((*_𝑟‘𝐹)‘𝑋)))
44	17	recnd 10669	. . . . 5 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ)
45	11	cjcld 14555	. . . . 5 ⊢ (𝜑 → (∗‘𝑋) ∈ ℂ)
46	11, 44, 45	mul32d 10850	. . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌)))
47	33, 43, 46	3eqtr2d 2862	. . 3 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌)))
48	47	fveq2d 6674	. 2 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))))
49		absval 14597	. . . 4 ⊢ (𝑋 ∈ ℂ → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋))))
50	11, 49	syl 17	. . 3 ⊢ (𝜑 → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋))))
51	50	oveq1d 7171	. 2 ⊢ (𝜑 → ((abs‘𝑋) · (√‘(𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌))))
52	24, 48, 51	3eqtr4d 2866	1 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = ((abs‘𝑋) · (√‘(𝑌 , 𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 · cmul 10542 ≤ cle 10676 ∗ccj 14455 √csqrt 14592 abscabs 14593 Basecbs 16483 ↾_s cress 16484 ._rcmulr 16566 *_𝑟cstv 16567 Scalarcsca 16568 ·_𝑠 cvsca 16569 ·_𝑖cip 16570 LModclmod 19634 ℂ_fldccnfld 20545 PreHilcphl 20768 ℂModcclm 23666 toℂPreHilctcph 23771

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-fz 12894 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-subg 18276 df-ghm 18356 df-cmn 18908 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-rnghom 19467 df-drng 19504 df-subrg 19533 df-staf 19616 df-srng 19617 df-lmod 19636 df-lmhm 19794 df-lvec 19875 df-sra 19944 df-rgmod 19945 df-cnfld 20546 df-phl 20770 df-clm 23667

This theorem is referenced by: tcphcph 23840

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