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Theorem tchcphlem2 23035

Description: Lemma for tchcph 23036: homogeneity. (Contributed by Mario Carneiro, 8-Oct-2015.)

**Hypotheses**
Ref	Expression
tchval.n	⊢ 𝐺 = (toℂHil‘𝑊)
tchcph.v	⊢ 𝑉 = (Base‘𝑊)
tchcph.f	⊢ 𝐹 = (Scalar‘𝑊)
tchcph.1	⊢ (𝜑 → 𝑊 ∈ PreHil)
tchcph.2	⊢ (𝜑 → 𝐹 = (ℂ_fld ↾_s 𝐾))
tchcph.h	⊢ , = (·_𝑖‘𝑊)
tchcph.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾)
tchcph.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
tchcph.k	⊢ 𝐾 = (Base‘𝐹)
tchcph.s	⊢ · = ( ·_𝑠 ‘𝑊)
tchcphlem2.3	⊢ (𝜑 → 𝑋 ∈ 𝐾)
tchcphlem2.4	⊢ (𝜑 → 𝑌 ∈ 𝑉)

**Assertion**
Ref	Expression
tchcphlem2	⊢ (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 tchcphlem2**
Step	Hyp	Ref	Expression
1		tchval.n	. . . . . . 7 ⊢ 𝐺 = (toℂHil‘𝑊)
2		tchcph.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
3		tchcph.f	. . . . . . 7 ⊢ 𝐹 = (Scalar‘𝑊)
4		tchcph.1	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ PreHil)
5		tchcph.2	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (ℂ_fld ↾_s 𝐾))
6	1, 2, 3, 4, 5	tchclm 23031	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂMod)
7		tchcph.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐹)
8	3, 7	clmsscn 22879	. . . . . 6 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ)
9	6, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ⊆ ℂ)
10		tchcphlem2.3	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐾)
11	9, 10	sseldd 3604	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ)
12	11	cjmulrcld 13946	. . 3 ⊢ (𝜑 → (𝑋 · (∗‘𝑋)) ∈ ℝ)
13	11	cjmulge0d 13948	. . 3 ⊢ (𝜑 → 0 ≤ (𝑋 · (∗‘𝑋)))
14		tchcphlem2.4	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15		tchcph.h	. . . . 5 ⊢ , = (·_𝑖‘𝑊)
16	1, 2, 3, 4, 5, 15	tchcphlem3 23032	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ)
17	14, 16	mpdan 702	. . 3 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ)
18		tchcph.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
19	18	ralrimiva 2966	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥))
20		oveq12 6659	. . . . . . 7 ⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌))
21	20	anidms 677	. . . . . 6 ⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌))
22	21	breq2d 4665	. . . . 5 ⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌)))
23	22	rspcv 3305	. . . 4 ⊢ (𝑌 ∈ 𝑉 → (∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥) → 0 ≤ (𝑌 , 𝑌)))
24	14, 19, 23	sylc 65	. . 3 ⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌))
25	12, 13, 17, 24	sqrtmuld 14163	. 2 ⊢ (𝜑 → (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌))))
26		phllmod 19975	. . . . . . 7 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
27	4, 26	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod)
28		tchcph.s	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝑊)
29	2, 3, 28, 7	lmodvscl 18880	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) ∈ 𝑉)
30	27, 10, 14, 29	syl3anc 1326	. . . . 5 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑉)
31		eqid 2622	. . . . . 6 ⊢ (._r‘𝐹) = (._r‘𝐹)
32		eqid 2622	. . . . . 6 ⊢ (_𝑟‘𝐹) = (_𝑟‘𝐹)
33	3, 15, 2, 7, 28, 31, 32	ipassr 19991	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ ((𝑋 · 𝑌) ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾)) → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = (((𝑋 · 𝑌) , 𝑌)(._r‘𝐹)((*_𝑟‘𝐹)‘𝑋)))
34	4, 30, 14, 10, 33	syl13anc 1328	. . . 4 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = (((𝑋 · 𝑌) , 𝑌)(._r‘𝐹)((*_𝑟‘𝐹)‘𝑋)))
35	3	clmmul 22875	. . . . . 6 ⊢ (𝑊 ∈ ℂMod → · = (._r‘𝐹))
36	6, 35	syl 17	. . . . 5 ⊢ (𝜑 → · = (._r‘𝐹))
37	36	oveqd 6667	. . . . . 6 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = (𝑋(._r‘𝐹)(𝑌 , 𝑌)))
38	3, 15, 2, 7, 28, 31	ipass 19990	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑌) , 𝑌) = (𝑋(._r‘𝐹)(𝑌 , 𝑌)))
39	4, 10, 14, 14, 38	syl13anc 1328	. . . . . 6 ⊢ (𝜑 → ((𝑋 · 𝑌) , 𝑌) = (𝑋(._r‘𝐹)(𝑌 , 𝑌)))
40	37, 39	eqtr4d 2659	. . . . 5 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = ((𝑋 · 𝑌) , 𝑌))
41	3	clmcj 22876	. . . . . . 7 ⊢ (𝑊 ∈ ℂMod → ∗ = (*_𝑟‘𝐹))
42	6, 41	syl 17	. . . . . 6 ⊢ (𝜑 → ∗ = (*_𝑟‘𝐹))
43	42	fveq1d 6193	. . . . 5 ⊢ (𝜑 → (∗‘𝑋) = ((*_𝑟‘𝐹)‘𝑋))
44	36, 40, 43	oveq123d 6671	. . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = (((𝑋 · 𝑌) , 𝑌)(._r‘𝐹)((*_𝑟‘𝐹)‘𝑋)))
45	17	recnd 10068	. . . . 5 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ)
46	11	cjcld 13936	. . . . 5 ⊢ (𝜑 → (∗‘𝑋) ∈ ℂ)
47	11, 45, 46	mul32d 10246	. . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌)))
48	34, 44, 47	3eqtr2d 2662	. . 3 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌)))
49	48	fveq2d 6195	. 2 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))))
50		absval 13978	. . . 4 ⊢ (𝑋 ∈ ℂ → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋))))
51	11, 50	syl 17	. . 3 ⊢ (𝜑 → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋))))
52	51	oveq1d 6665	. 2 ⊢ (𝜑 → ((abs‘𝑋) · (√‘(𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌))))
53	25, 49, 52	3eqtr4d 2666	1 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = ((abs‘𝑋) · (√‘(𝑌 , 𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 · cmul 9941 ≤ cle 10075 ∗ccj 13836 √csqrt 13973 abscabs 13974 Basecbs 15857 ↾_s cress 15858 ._rcmulr 15942 *_𝑟cstv 15943 Scalarcsca 15944 ·_𝑠 cvsca 15945 ·_𝑖cip 15946 LModclmod 18863 ℂ_fldccnfld 19746 PreHilcphl 19969 ℂModcclm 22862 toℂHilctch 22967

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-grp 17425 df-subg 17591 df-ghm 17658 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-rnghom 18715 df-drng 18749 df-subrg 18778 df-staf 18845 df-srng 18846 df-lmod 18865 df-lmhm 19022 df-lvec 19103 df-sra 19172 df-rgmod 19173 df-cnfld 19747 df-phl 19971 df-clm 22863

This theorem is referenced by: tchcph 23036

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