tcphcphlem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > tcphcphlem2		Structured version Visualization version GIF version

Theorem tcphcphlem2 25152

Description: Lemma for tcphcph 25153: 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 25148	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂMod)
7		tcphcph.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐹)
8	3, 7	clmsscn 24995	. . . . . 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 3938	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ)
12	11	cjmulrcld 15131	. . 3 ⊢ (𝜑 → (𝑋 · (∗‘𝑋)) ∈ ℝ)
13	11	cjmulge0d 15133	. . 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 25149	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ)
17	14, 16	mpdan 687	. . 3 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ)
18		oveq12 7362	. . . . . 6 ⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌))
19	18	anidms 566	. . . . 5 ⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌))
20	19	breq2d 5107	. . . 4 ⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌)))
21		tcphcph.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
22	21	ralrimiva 3121	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥))
23	20, 22, 14	rspcdva 3580	. . 3 ⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌))
24	12, 13, 17, 23	sqrtmuld 15350	. 2 ⊢ (𝜑 → (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌))))
25		phllmod 21555	. . . . . . 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 20799	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) ∈ 𝑉)
29	26, 10, 14, 28	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑉)
30		eqid 2729	. . . . . 6 ⊢ (._r‘𝐹) = (._r‘𝐹)
31		eqid 2729	. . . . . 6 ⊢ (_𝑟‘𝐹) = (_𝑟‘𝐹)
32	3, 15, 2, 7, 27, 30, 31	ipassr 21571	. . . . 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 1374	. . . 4 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = (((𝑋 · 𝑌) , 𝑌)(._r‘𝐹)((*_𝑟‘𝐹)‘𝑋)))
34	3	clmmul 24991	. . . . . 6 ⊢ (𝑊 ∈ ℂMod → · = (._r‘𝐹))
35	6, 34	syl 17	. . . . 5 ⊢ (𝜑 → · = (._r‘𝐹))
36	35	oveqd 7370	. . . . . 6 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = (𝑋(._r‘𝐹)(𝑌 , 𝑌)))
37	3, 15, 2, 7, 27, 30	ipass 21570	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑌) , 𝑌) = (𝑋(._r‘𝐹)(𝑌 , 𝑌)))
38	4, 10, 14, 14, 37	syl13anc 1374	. . . . . 6 ⊢ (𝜑 → ((𝑋 · 𝑌) , 𝑌) = (𝑋(._r‘𝐹)(𝑌 , 𝑌)))
39	36, 38	eqtr4d 2767	. . . . 5 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = ((𝑋 · 𝑌) , 𝑌))
40	3	clmcj 24992	. . . . . . 7 ⊢ (𝑊 ∈ ℂMod → ∗ = (*_𝑟‘𝐹))
41	6, 40	syl 17	. . . . . 6 ⊢ (𝜑 → ∗ = (*_𝑟‘𝐹))
42	41	fveq1d 6828	. . . . 5 ⊢ (𝜑 → (∗‘𝑋) = ((*_𝑟‘𝐹)‘𝑋))
43	35, 39, 42	oveq123d 7374	. . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = (((𝑋 · 𝑌) , 𝑌)(._r‘𝐹)((*_𝑟‘𝐹)‘𝑋)))
44	17	recnd 11162	. . . . 5 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ)
45	11	cjcld 15121	. . . . 5 ⊢ (𝜑 → (∗‘𝑋) ∈ ℂ)
46	11, 44, 45	mul32d 11344	. . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌)))
47	33, 43, 46	3eqtr2d 2770	. . 3 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌)))
48	47	fveq2d 6830	. 2 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))))
49		absval 15163	. . . 4 ⊢ (𝑋 ∈ ℂ → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋))))
50	11, 49	syl 17	. . 3 ⊢ (𝜑 → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋))))
51	50	oveq1d 7368	. 2 ⊢ (𝜑 → ((abs‘𝑋) · (√‘(𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌))))
52	24, 48, 51	3eqtr4d 2774	1 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = ((abs‘𝑋) · (√‘(𝑌 , 𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ⊆ wss 3905 class class class wbr 5095 ‘cfv 6486 (class class class)co 7353 ℂcc 11026 ℝcr 11027 0cc0 11028 · cmul 11033 ≤ cle 11169 ∗ccj 15021 √csqrt 15158 abscabs 15159 Basecbs 17138 ↾_s cress 17159 ._rcmulr 17180 *_𝑟cstv 17181 Scalarcsca 17182 ·_𝑠 cvsca 17183 ·_𝑖cip 17184 LModclmod 20781 ℂ_fldccnfld 21279 PreHilcphl 21549 ℂModcclm 24978 toℂPreHilctcph 25083

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107 ax-mulf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9351 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-rp 12912 df-fz 13429 df-seq 13927 df-exp 13987 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-0g 17363 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-mhm 18675 df-grp 18833 df-minusg 18834 df-subg 19020 df-ghm 19110 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-oppr 20240 df-dvdsr 20260 df-unit 20261 df-rhm 20375 df-subrg 20473 df-drng 20634 df-staf 20742 df-srng 20743 df-lmod 20783 df-lmhm 20944 df-lvec 21025 df-sra 21095 df-rgmod 21096 df-cnfld 21280 df-phl 21551 df-clm 24979

This theorem is referenced by: tcphcph 25153

W3C validator