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Mirrors > Home > MPE Home > Th. List > tcphcphlem2 | Structured version Visualization version GIF version |
Description: Lemma for tcphcph 24499: homogeneity. (Contributed by Mario Carneiro, 8-Oct-2015.) |
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 = (ℂfld ↾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) |
Ref | Expression |
---|---|
tcphcphlem2 | ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = ((abs‘𝑋) · (√‘(𝑌 , 𝑌)))) |
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 = (ℂfld ↾s 𝐾)) | |
6 | 1, 2, 3, 4, 5 | phclm 24494 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
7 | tcphcph.k | . . . . . . 7 ⊢ 𝐾 = (Base‘𝐹) | |
8 | 3, 7 | clmsscn 24340 | . . . . . 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 3932 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
12 | 11 | cjmulrcld 15008 | . . 3 ⊢ (𝜑 → (𝑋 · (∗‘𝑋)) ∈ ℝ) |
13 | 11 | cjmulge0d 15010 | . . 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 24495 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ) |
17 | 14, 16 | mpdan 684 | . . 3 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ) |
18 | oveq12 7338 | . . . . . 6 ⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌)) | |
19 | 18 | anidms 567 | . . . . 5 ⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌)) |
20 | 19 | breq2d 5101 | . . . 4 ⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌))) |
21 | tcphcph.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) | |
22 | 21 | ralrimiva 3139 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥)) |
23 | 20, 22, 14 | rspcdva 3571 | . . 3 ⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌)) |
24 | 12, 13, 17, 23 | sqrtmuld 15227 | . 2 ⊢ (𝜑 → (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌)))) |
25 | phllmod 20933 | . . . . . . 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 20238 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) ∈ 𝑉) |
29 | 26, 10, 14, 28 | syl3anc 1370 | . . . . 5 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑉) |
30 | eqid 2736 | . . . . . 6 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
31 | eqid 2736 | . . . . . 6 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹) | |
32 | 3, 15, 2, 7, 27, 30, 31 | ipassr 20949 | . . . . 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 1371 | . . . 4 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = (((𝑋 · 𝑌) , 𝑌)(.r‘𝐹)((*𝑟‘𝐹)‘𝑋))) |
34 | 3 | clmmul 24336 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → · = (.r‘𝐹)) |
35 | 6, 34 | syl 17 | . . . . 5 ⊢ (𝜑 → · = (.r‘𝐹)) |
36 | 35 | oveqd 7346 | . . . . . 6 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = (𝑋(.r‘𝐹)(𝑌 , 𝑌))) |
37 | 3, 15, 2, 7, 27, 30 | ipass 20948 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑌) , 𝑌) = (𝑋(.r‘𝐹)(𝑌 , 𝑌))) |
38 | 4, 10, 14, 14, 37 | syl13anc 1371 | . . . . . 6 ⊢ (𝜑 → ((𝑋 · 𝑌) , 𝑌) = (𝑋(.r‘𝐹)(𝑌 , 𝑌))) |
39 | 36, 38 | eqtr4d 2779 | . . . . 5 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = ((𝑋 · 𝑌) , 𝑌)) |
40 | 3 | clmcj 24337 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → ∗ = (*𝑟‘𝐹)) |
41 | 6, 40 | syl 17 | . . . . . 6 ⊢ (𝜑 → ∗ = (*𝑟‘𝐹)) |
42 | 41 | fveq1d 6821 | . . . . 5 ⊢ (𝜑 → (∗‘𝑋) = ((*𝑟‘𝐹)‘𝑋)) |
43 | 35, 39, 42 | oveq123d 7350 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = (((𝑋 · 𝑌) , 𝑌)(.r‘𝐹)((*𝑟‘𝐹)‘𝑋))) |
44 | 17 | recnd 11096 | . . . . 5 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ) |
45 | 11 | cjcld 14998 | . . . . 5 ⊢ (𝜑 → (∗‘𝑋) ∈ ℂ) |
46 | 11, 44, 45 | mul32d 11278 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) |
47 | 33, 43, 46 | 3eqtr2d 2782 | . . 3 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) |
48 | 47 | fveq2d 6823 | . 2 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌)))) |
49 | absval 15040 | . . . 4 ⊢ (𝑋 ∈ ℂ → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋)))) | |
50 | 11, 49 | syl 17 | . . 3 ⊢ (𝜑 → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋)))) |
51 | 50 | oveq1d 7344 | . 2 ⊢ (𝜑 → ((abs‘𝑋) · (√‘(𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌)))) |
52 | 24, 48, 51 | 3eqtr4d 2786 | 1 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = ((abs‘𝑋) · (√‘(𝑌 , 𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ⊆ wss 3897 class class class wbr 5089 ‘cfv 6473 (class class class)co 7329 ℂcc 10962 ℝcr 10963 0cc0 10964 · cmul 10969 ≤ cle 11103 ∗ccj 14898 √csqrt 15035 abscabs 15036 Basecbs 17001 ↾s cress 17030 .rcmulr 17052 *𝑟cstv 17053 Scalarcsca 17054 ·𝑠 cvsca 17055 ·𝑖cip 17056 LModclmod 20221 ℂfldccnfld 20695 PreHilcphl 20927 ℂModcclm 24323 toℂPreHilctcph 24429 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-pre-sup 11042 ax-addf 11043 ax-mulf 11044 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4852 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-om 7773 df-1st 7891 df-2nd 7892 df-tpos 8104 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-sup 9291 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-div 11726 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-7 12134 df-8 12135 df-9 12136 df-n0 12327 df-z 12413 df-dec 12531 df-uz 12676 df-rp 12824 df-fz 13333 df-seq 13815 df-exp 13876 df-cj 14901 df-re 14902 df-im 14903 df-sqrt 15037 df-abs 15038 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-starv 17066 df-sca 17067 df-vsca 17068 df-ip 17069 df-tset 17070 df-ple 17071 df-ds 17073 df-unif 17074 df-0g 17241 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-mhm 18519 df-grp 18668 df-subg 18840 df-ghm 18920 df-cmn 19475 df-mgp 19808 df-ur 19825 df-ring 19872 df-cring 19873 df-oppr 19949 df-dvdsr 19970 df-unit 19971 df-rnghom 20046 df-drng 20087 df-subrg 20119 df-staf 20203 df-srng 20204 df-lmod 20223 df-lmhm 20382 df-lvec 20463 df-sra 20532 df-rgmod 20533 df-cnfld 20696 df-phl 20929 df-clm 24324 |
This theorem is referenced by: tcphcph 24499 |
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