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| Mirrors > Home > MPE Home > Th. List > tcphcphlem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for tcphcph 25271: 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 25266 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
| 7 | tcphcph.k | . . . . . . 7 ⊢ 𝐾 = (Base‘𝐹) | |
| 8 | 3, 7 | clmsscn 25112 | . . . . . 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 3984 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
| 12 | 11 | cjmulrcld 15245 | . . 3 ⊢ (𝜑 → (𝑋 · (∗‘𝑋)) ∈ ℝ) |
| 13 | 11 | cjmulge0d 15247 | . . 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 25267 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ) |
| 17 | 14, 16 | mpdan 687 | . . 3 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ) |
| 18 | oveq12 7440 | . . . . . 6 ⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌)) | |
| 19 | 18 | anidms 566 | . . . . 5 ⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌)) |
| 20 | 19 | breq2d 5155 | . . . 4 ⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌))) |
| 21 | tcphcph.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) | |
| 22 | 21 | ralrimiva 3146 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥)) |
| 23 | 20, 22, 14 | rspcdva 3623 | . . 3 ⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌)) |
| 24 | 12, 13, 17, 23 | sqrtmuld 15463 | . 2 ⊢ (𝜑 → (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌)))) |
| 25 | phllmod 21648 | . . . . . . 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 20876 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) ∈ 𝑉) |
| 29 | 26, 10, 14, 28 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑉) |
| 30 | eqid 2737 | . . . . . 6 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
| 31 | eqid 2737 | . . . . . 6 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹) | |
| 32 | 3, 15, 2, 7, 27, 30, 31 | ipassr 21664 | . . . . 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 25108 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → · = (.r‘𝐹)) |
| 35 | 6, 34 | syl 17 | . . . . 5 ⊢ (𝜑 → · = (.r‘𝐹)) |
| 36 | 35 | oveqd 7448 | . . . . . 6 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = (𝑋(.r‘𝐹)(𝑌 , 𝑌))) |
| 37 | 3, 15, 2, 7, 27, 30 | ipass 21663 | . . . . . . 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 2780 | . . . . 5 ⊢ (𝜑 → (𝑋 · (𝑌 , 𝑌)) = ((𝑋 · 𝑌) , 𝑌)) |
| 40 | 3 | clmcj 25109 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → ∗ = (*𝑟‘𝐹)) |
| 41 | 6, 40 | syl 17 | . . . . . 6 ⊢ (𝜑 → ∗ = (*𝑟‘𝐹)) |
| 42 | 41 | fveq1d 6908 | . . . . 5 ⊢ (𝜑 → (∗‘𝑋) = ((*𝑟‘𝐹)‘𝑋)) |
| 43 | 35, 39, 42 | oveq123d 7452 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = (((𝑋 · 𝑌) , 𝑌)(.r‘𝐹)((*𝑟‘𝐹)‘𝑋))) |
| 44 | 17 | recnd 11289 | . . . . 5 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ) |
| 45 | 11 | cjcld 15235 | . . . . 5 ⊢ (𝜑 → (∗‘𝑋) ∈ ℂ) |
| 46 | 11, 44, 45 | mul32d 11471 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝑌 , 𝑌)) · (∗‘𝑋)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) |
| 47 | 33, 43, 46 | 3eqtr2d 2783 | . . 3 ⊢ (𝜑 → ((𝑋 · 𝑌) , (𝑋 · 𝑌)) = ((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌))) |
| 48 | 47 | fveq2d 6910 | . 2 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = (√‘((𝑋 · (∗‘𝑋)) · (𝑌 , 𝑌)))) |
| 49 | absval 15277 | . . . 4 ⊢ (𝑋 ∈ ℂ → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋)))) | |
| 50 | 11, 49 | syl 17 | . . 3 ⊢ (𝜑 → (abs‘𝑋) = (√‘(𝑋 · (∗‘𝑋)))) |
| 51 | 50 | oveq1d 7446 | . 2 ⊢ (𝜑 → ((abs‘𝑋) · (√‘(𝑌 , 𝑌))) = ((√‘(𝑋 · (∗‘𝑋))) · (√‘(𝑌 , 𝑌)))) |
| 52 | 24, 48, 51 | 3eqtr4d 2787 | 1 ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = ((abs‘𝑋) · (√‘(𝑌 , 𝑌)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 · cmul 11160 ≤ cle 11296 ∗ccj 15135 √csqrt 15272 abscabs 15273 Basecbs 17247 ↾s cress 17274 .rcmulr 17298 *𝑟cstv 17299 Scalarcsca 17300 ·𝑠 cvsca 17301 ·𝑖cip 17302 LModclmod 20858 ℂfldccnfld 21364 PreHilcphl 21642 ℂModcclm 25095 toℂPreHilctcph 25201 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-minusg 18955 df-subg 19141 df-ghm 19231 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-rhm 20472 df-subrg 20570 df-drng 20731 df-staf 20840 df-srng 20841 df-lmod 20860 df-lmhm 21021 df-lvec 21102 df-sra 21172 df-rgmod 21173 df-cnfld 21365 df-phl 21644 df-clm 25096 |
| This theorem is referenced by: tcphcph 25271 |
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