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Mirrors > Home > MPE Home > Th. List > tcphcphlem2 | Structured version Visualization version GIF version |
Description: Lemma for tcphcph 23833: 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 23828 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
7 | tcphcph.k | . . . . . . 7 ⊢ 𝐾 = (Base‘𝐹) | |
8 | 3, 7 | clmsscn 23676 | . . . . . 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 3961 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
12 | 11 | cjmulrcld 14558 | . . 3 ⊢ (𝜑 → (𝑋 · (∗‘𝑋)) ∈ ℝ) |
13 | 11 | cjmulge0d 14560 | . . 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 23829 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ) |
17 | 14, 16 | mpdan 685 | . . 3 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ) |
18 | oveq12 7158 | . . . . . 6 ⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌)) | |
19 | 18 | anidms 569 | . . . . 5 ⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌)) |
20 | 19 | breq2d 5071 | . . . 4 ⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌))) |
21 | tcphcph.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) | |
22 | 21 | ralrimiva 3181 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥)) |
23 | 20, 22, 14 | rspcdva 3622 | . . 3 ⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌)) |
24 | 12, 13, 17, 23 | sqrtmuld 14777 | . 2 ⊢ (𝜑 → (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌)))) |
25 | phllmod 20767 | . . . . . . 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 19644 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) ∈ 𝑉) |
29 | 26, 10, 14, 28 | syl3anc 1366 | . . . . 5 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑉) |
30 | eqid 2820 | . . . . . 6 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
31 | eqid 2820 | . . . . . 6 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹) | |
32 | 3, 15, 2, 7, 27, 30, 31 | ipassr 20783 | . . . . 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 1367 | . . . 4 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = (((𝑋 · 𝑌) , 𝑌)(.r‘𝐹)((*𝑟‘𝐹)‘𝑋))) |
34 | 3 | clmmul 23672 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → · = (.r‘𝐹)) |
35 | 6, 34 | syl 17 | . . . . 5 ⊢ (𝜑 → · = (.r‘𝐹)) |
36 | 35 | oveqd 7166 | . . . . . 6 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = (𝑋(.r‘𝐹)(𝑌 , 𝑌))) |
37 | 3, 15, 2, 7, 27, 30 | ipass 20782 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑌) , 𝑌) = (𝑋(.r‘𝐹)(𝑌 , 𝑌))) |
38 | 4, 10, 14, 14, 37 | syl13anc 1367 | . . . . . 6 ⊢ (𝜑 → ((𝑋 · 𝑌) , 𝑌) = (𝑋(.r‘𝐹)(𝑌 , 𝑌))) |
39 | 36, 38 | eqtr4d 2858 | . . . . 5 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = ((𝑋 · 𝑌) , 𝑌)) |
40 | 3 | clmcj 23673 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → ∗ = (*𝑟‘𝐹)) |
41 | 6, 40 | syl 17 | . . . . . 6 ⊢ (𝜑 → ∗ = (*𝑟‘𝐹)) |
42 | 41 | fveq1d 6665 | . . . . 5 ⊢ (𝜑 → (∗‘𝑋) = ((*𝑟‘𝐹)‘𝑋)) |
43 | 35, 39, 42 | oveq123d 7170 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = (((𝑋 · 𝑌) , 𝑌)(.r‘𝐹)((*𝑟‘𝐹)‘𝑋))) |
44 | 17 | recnd 10662 | . . . . 5 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ) |
45 | 11 | cjcld 14548 | . . . . 5 ⊢ (𝜑 → (∗‘𝑋) ∈ ℂ) |
46 | 11, 44, 45 | mul32d 10843 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) |
47 | 33, 43, 46 | 3eqtr2d 2861 | . . 3 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) |
48 | 47 | fveq2d 6667 | . 2 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌)))) |
49 | absval 14590 | . . . 4 ⊢ (𝑋 ∈ ℂ → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋)))) | |
50 | 11, 49 | syl 17 | . . 3 ⊢ (𝜑 → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋)))) |
51 | 50 | oveq1d 7164 | . 2 ⊢ (𝜑 → ((abs‘𝑋) · (√‘(𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌)))) |
52 | 24, 48, 51 | 3eqtr4d 2865 | 1 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = ((abs‘𝑋) · (√‘(𝑌 , 𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ⊆ wss 3929 class class class wbr 5059 ‘cfv 6348 (class class class)co 7149 ℂcc 10528 ℝcr 10529 0cc0 10530 · cmul 10535 ≤ cle 10669 ∗ccj 14448 √csqrt 14585 abscabs 14586 Basecbs 16476 ↾s cress 16477 .rcmulr 16559 *𝑟cstv 16560 Scalarcsca 16561 ·𝑠 cvsca 16562 ·𝑖cip 16563 LModclmod 19627 ℂfldccnfld 20538 PreHilcphl 20761 ℂModcclm 23659 toℂPreHilctcph 23764 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-addf 10609 ax-mulf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12890 df-seq 13367 df-exp 13427 df-cj 14451 df-re 14452 df-im 14453 df-sqrt 14587 df-abs 14588 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-mulr 16572 df-starv 16573 df-sca 16574 df-vsca 16575 df-ip 16576 df-tset 16577 df-ple 16578 df-ds 16580 df-unif 16581 df-0g 16708 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-mhm 17949 df-grp 18099 df-subg 18269 df-ghm 18349 df-cmn 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19366 df-dvdsr 19384 df-unit 19385 df-rnghom 19460 df-drng 19497 df-subrg 19526 df-staf 19609 df-srng 19610 df-lmod 19629 df-lmhm 19787 df-lvec 19868 df-sra 19937 df-rgmod 19938 df-cnfld 20539 df-phl 20763 df-clm 23660 |
This theorem is referenced by: tcphcph 23833 |
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