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Mirrors > Home > MPE Home > Th. List > tcphcphlem2 | Structured version Visualization version GIF version |
Description: Lemma for tcphcph 23406: 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 23401 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
7 | tcphcph.k | . . . . . . 7 ⊢ 𝐾 = (Base‘𝐹) | |
8 | 3, 7 | clmsscn 23249 | . . . . . 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 3829 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
12 | 11 | cjmulrcld 14324 | . . 3 ⊢ (𝜑 → (𝑋 · (∗‘𝑋)) ∈ ℝ) |
13 | 11 | cjmulge0d 14326 | . . 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 23402 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ) |
17 | 14, 16 | mpdan 680 | . . 3 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ) |
18 | oveq12 6915 | . . . . . 6 ⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌)) | |
19 | 18 | anidms 564 | . . . . 5 ⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌)) |
20 | 19 | breq2d 4886 | . . . 4 ⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌))) |
21 | tcphcph.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) | |
22 | 21 | ralrimiva 3176 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥)) |
23 | 20, 22, 14 | rspcdva 3533 | . . 3 ⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌)) |
24 | 12, 13, 17, 23 | sqrtmuld 14541 | . 2 ⊢ (𝜑 → (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌)))) |
25 | phllmod 20338 | . . . . . . 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 19237 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) ∈ 𝑉) |
29 | 26, 10, 14, 28 | syl3anc 1496 | . . . . 5 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑉) |
30 | eqid 2826 | . . . . . 6 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
31 | eqid 2826 | . . . . . 6 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹) | |
32 | 3, 15, 2, 7, 27, 30, 31 | ipassr 20354 | . . . . 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 1497 | . . . 4 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = (((𝑋 · 𝑌) , 𝑌)(.r‘𝐹)((*𝑟‘𝐹)‘𝑋))) |
34 | 3 | clmmul 23245 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → · = (.r‘𝐹)) |
35 | 6, 34 | syl 17 | . . . . 5 ⊢ (𝜑 → · = (.r‘𝐹)) |
36 | 35 | oveqd 6923 | . . . . . 6 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = (𝑋(.r‘𝐹)(𝑌 , 𝑌))) |
37 | 3, 15, 2, 7, 27, 30 | ipass 20353 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑌) , 𝑌) = (𝑋(.r‘𝐹)(𝑌 , 𝑌))) |
38 | 4, 10, 14, 14, 37 | syl13anc 1497 | . . . . . 6 ⊢ (𝜑 → ((𝑋 · 𝑌) , 𝑌) = (𝑋(.r‘𝐹)(𝑌 , 𝑌))) |
39 | 36, 38 | eqtr4d 2865 | . . . . 5 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = ((𝑋 · 𝑌) , 𝑌)) |
40 | 3 | clmcj 23246 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → ∗ = (*𝑟‘𝐹)) |
41 | 6, 40 | syl 17 | . . . . . 6 ⊢ (𝜑 → ∗ = (*𝑟‘𝐹)) |
42 | 41 | fveq1d 6436 | . . . . 5 ⊢ (𝜑 → (∗‘𝑋) = ((*𝑟‘𝐹)‘𝑋)) |
43 | 35, 39, 42 | oveq123d 6927 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = (((𝑋 · 𝑌) , 𝑌)(.r‘𝐹)((*𝑟‘𝐹)‘𝑋))) |
44 | 17 | recnd 10386 | . . . . 5 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ) |
45 | 11 | cjcld 14314 | . . . . 5 ⊢ (𝜑 → (∗‘𝑋) ∈ ℂ) |
46 | 11, 44, 45 | mul32d 10566 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) |
47 | 33, 43, 46 | 3eqtr2d 2868 | . . 3 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) |
48 | 47 | fveq2d 6438 | . 2 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌)))) |
49 | absval 14356 | . . . 4 ⊢ (𝑋 ∈ ℂ → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋)))) | |
50 | 11, 49 | syl 17 | . . 3 ⊢ (𝜑 → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋)))) |
51 | 50 | oveq1d 6921 | . 2 ⊢ (𝜑 → ((abs‘𝑋) · (√‘(𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌)))) |
52 | 24, 48, 51 | 3eqtr4d 2872 | 1 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = ((abs‘𝑋) · (√‘(𝑌 , 𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ⊆ wss 3799 class class class wbr 4874 ‘cfv 6124 (class class class)co 6906 ℂcc 10251 ℝcr 10252 0cc0 10253 · cmul 10258 ≤ cle 10393 ∗ccj 14214 √csqrt 14351 abscabs 14352 Basecbs 16223 ↾s cress 16224 .rcmulr 16307 *𝑟cstv 16308 Scalarcsca 16309 ·𝑠 cvsca 16310 ·𝑖cip 16311 LModclmod 19220 ℂfldccnfld 20107 PreHilcphl 20332 ℂModcclm 23232 toℂPreHilctcph 23337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 ax-addf 10332 ax-mulf 10333 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-tpos 7618 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-map 8125 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-sup 8618 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-z 11706 df-dec 11823 df-uz 11970 df-rp 12114 df-fz 12621 df-seq 13097 df-exp 13156 df-cj 14217 df-re 14218 df-im 14219 df-sqrt 14353 df-abs 14354 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-starv 16321 df-sca 16322 df-vsca 16323 df-ip 16324 df-tset 16325 df-ple 16326 df-ds 16328 df-unif 16329 df-0g 16456 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-mhm 17689 df-grp 17780 df-subg 17943 df-ghm 18010 df-cmn 18549 df-mgp 18845 df-ur 18857 df-ring 18904 df-cring 18905 df-oppr 18978 df-dvdsr 18996 df-unit 18997 df-rnghom 19072 df-drng 19106 df-subrg 19135 df-staf 19202 df-srng 19203 df-lmod 19222 df-lmhm 19382 df-lvec 19463 df-sra 19534 df-rgmod 19535 df-cnfld 20108 df-phl 20334 df-clm 23233 |
This theorem is referenced by: tcphcph 23406 |
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