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Mirrors > Home > MPE Home > Th. List > tchcphlem2 | Structured version Visualization version GIF version |
Description: Lemma for tchcph 22844: homogeneity. (Contributed by Mario Carneiro, 8-Oct-2015.) |
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 = (ℂfld ↾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) |
Ref | Expression |
---|---|
tchcphlem2 | ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = ((abs‘𝑋) · (√‘(𝑌 , 𝑌)))) |
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 = (ℂfld ↾s 𝐾)) | |
6 | 1, 2, 3, 4, 5 | tchclm 22839 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
7 | tchcph.k | . . . . . . 7 ⊢ 𝐾 = (Base‘𝐹) | |
8 | 3, 7 | clmsscn 22687 | . . . . . 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 3569 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
12 | 11 | cjmulrcld 13794 | . . 3 ⊢ (𝜑 → (𝑋 · (∗‘𝑋)) ∈ ℝ) |
13 | 11 | cjmulge0d 13796 | . . 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 22840 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ) |
17 | 14, 16 | mpdan 699 | . . 3 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ) |
18 | tchcph.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) | |
19 | 18 | ralrimiva 2949 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥)) |
20 | oveq12 6558 | . . . . . . 7 ⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌)) | |
21 | 20 | anidms 675 | . . . . . 6 ⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌)) |
22 | 21 | breq2d 4595 | . . . . 5 ⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌))) |
23 | 22 | rspcv 3278 | . . . 4 ⊢ (𝑌 ∈ 𝑉 → (∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥) → 0 ≤ (𝑌 , 𝑌))) |
24 | 14, 19, 23 | sylc 63 | . . 3 ⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌)) |
25 | 12, 13, 17, 24 | sqrtmuld 14011 | . 2 ⊢ (𝜑 → (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌)))) |
26 | phllmod 19794 | . . . . . . 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 18703 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) ∈ 𝑉) |
30 | 27, 10, 14, 29 | syl3anc 1318 | . . . . 5 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑉) |
31 | eqid 2610 | . . . . . 6 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
32 | eqid 2610 | . . . . . 6 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹) | |
33 | 3, 15, 2, 7, 28, 31, 32 | ipassr 19810 | . . . . 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 1320 | . . . 4 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = (((𝑋 · 𝑌) , 𝑌)(.r‘𝐹)((*𝑟‘𝐹)‘𝑋))) |
35 | 3 | clmmul 22683 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → · = (.r‘𝐹)) |
36 | 6, 35 | syl 17 | . . . . 5 ⊢ (𝜑 → · = (.r‘𝐹)) |
37 | 36 | oveqd 6566 | . . . . . 6 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = (𝑋(.r‘𝐹)(𝑌 , 𝑌))) |
38 | 3, 15, 2, 7, 28, 31 | ipass 19809 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑌) , 𝑌) = (𝑋(.r‘𝐹)(𝑌 , 𝑌))) |
39 | 4, 10, 14, 14, 38 | syl13anc 1320 | . . . . . 6 ⊢ (𝜑 → ((𝑋 · 𝑌) , 𝑌) = (𝑋(.r‘𝐹)(𝑌 , 𝑌))) |
40 | 37, 39 | eqtr4d 2647 | . . . . 5 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = ((𝑋 · 𝑌) , 𝑌)) |
41 | 3 | clmcj 22684 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → ∗ = (*𝑟‘𝐹)) |
42 | 6, 41 | syl 17 | . . . . . 6 ⊢ (𝜑 → ∗ = (*𝑟‘𝐹)) |
43 | 42 | fveq1d 6105 | . . . . 5 ⊢ (𝜑 → (∗‘𝑋) = ((*𝑟‘𝐹)‘𝑋)) |
44 | 36, 40, 43 | oveq123d 6570 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = (((𝑋 · 𝑌) , 𝑌)(.r‘𝐹)((*𝑟‘𝐹)‘𝑋))) |
45 | 17 | recnd 9947 | . . . . 5 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ) |
46 | 11 | cjcld 13784 | . . . . 5 ⊢ (𝜑 → (∗‘𝑋) ∈ ℂ) |
47 | 11, 45, 46 | mul32d 10125 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) |
48 | 34, 44, 47 | 3eqtr2d 2650 | . . 3 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) |
49 | 48 | fveq2d 6107 | . 2 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌)))) |
50 | absval 13826 | . . . 4 ⊢ (𝑋 ∈ ℂ → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋)))) | |
51 | 11, 50 | syl 17 | . . 3 ⊢ (𝜑 → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋)))) |
52 | 51 | oveq1d 6564 | . 2 ⊢ (𝜑 → ((abs‘𝑋) · (√‘(𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌)))) |
53 | 25, 49, 52 | 3eqtr4d 2654 | 1 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = ((abs‘𝑋) · (√‘(𝑌 , 𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 · cmul 9820 ≤ cle 9954 ∗ccj 13684 √csqrt 13821 abscabs 13822 Basecbs 15695 ↾s cress 15696 .rcmulr 15769 *𝑟cstv 15770 Scalarcsca 15771 ·𝑠 cvsca 15772 ·𝑖cip 15773 LModclmod 18686 ℂfldccnfld 19567 PreHilcphl 19788 ℂModcclm 22670 toℂHilctch 22775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-subg 17414 df-ghm 17481 df-cmn 18018 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-rnghom 18538 df-drng 18572 df-subrg 18601 df-staf 18668 df-srng 18669 df-lmod 18688 df-lmhm 18843 df-lvec 18924 df-sra 18993 df-rgmod 18994 df-cnfld 19568 df-phl 19790 df-clm 22671 |
This theorem is referenced by: tchcph 22844 |
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