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| Mirrors > Home > MPE Home > Th. List > cph2ass | Structured version Visualization version GIF version | ||
| Description: Move scalar multiplication to outside of inner product. See his35 31117. (Contributed by Mario Carneiro, 17-Oct-2015.) |
| Ref | Expression |
|---|---|
| cphipcj.h | ⊢ , = (·𝑖‘𝑊) |
| cphipcj.v | ⊢ 𝑉 = (Base‘𝑊) |
| cphass.f | ⊢ 𝐹 = (Scalar‘𝑊) |
| cphass.k | ⊢ 𝐾 = (Base‘𝐹) |
| cphass.s | ⊢ · = ( ·𝑠 ‘𝑊) |
| Ref | Expression |
|---|---|
| cph2ass | ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 · 𝐶) , (𝐵 · 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 , 𝐷))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1 1135 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑊 ∈ ℂPreHil) | |
| 2 | simp2r 1199 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐵 ∈ 𝐾) | |
| 3 | simp3l 1200 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
| 4 | simp3r 1201 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐷 ∈ 𝑉) | |
| 5 | cphipcj.h | . . . . 5 ⊢ , = (·𝑖‘𝑊) | |
| 6 | cphipcj.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
| 7 | cphass.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 8 | cphass.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
| 9 | cphass.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
| 10 | 5, 6, 7, 8, 9 | cphassr 25260 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐵 ∈ 𝐾 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , (𝐵 · 𝐷)) = ((∗‘𝐵) · (𝐶 , 𝐷))) |
| 11 | 1, 2, 3, 4, 10 | syl13anc 1371 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , (𝐵 · 𝐷)) = ((∗‘𝐵) · (𝐶 , 𝐷))) |
| 12 | 11 | oveq2d 7447 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐴 · (𝐶 , (𝐵 · 𝐷))) = (𝐴 · ((∗‘𝐵) · (𝐶 , 𝐷)))) |
| 13 | simp2l 1198 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐴 ∈ 𝐾) | |
| 14 | cphlmod 25222 | . . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod) | |
| 15 | 14 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑊 ∈ LMod) |
| 16 | 6, 7, 9, 8 | lmodvscl 20893 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐾 ∧ 𝐷 ∈ 𝑉) → (𝐵 · 𝐷) ∈ 𝑉) |
| 17 | 15, 2, 4, 16 | syl3anc 1370 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐵 · 𝐷) ∈ 𝑉) |
| 18 | 5, 6, 7, 8, 9 | cphass 25259 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐶 ∈ 𝑉 ∧ (𝐵 · 𝐷) ∈ 𝑉)) → ((𝐴 · 𝐶) , (𝐵 · 𝐷)) = (𝐴 · (𝐶 , (𝐵 · 𝐷)))) |
| 19 | 1, 13, 3, 17, 18 | syl13anc 1371 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 · 𝐶) , (𝐵 · 𝐷)) = (𝐴 · (𝐶 , (𝐵 · 𝐷)))) |
| 20 | cphclm 25237 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod) | |
| 21 | 20 | 3ad2ant1 1132 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑊 ∈ ℂMod) |
| 22 | 7, 8 | clmsscn 25126 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ) |
| 23 | 21, 22 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐾 ⊆ ℂ) |
| 24 | 23, 13 | sseldd 3996 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐴 ∈ ℂ) |
| 25 | 23, 2 | sseldd 3996 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐵 ∈ ℂ) |
| 26 | 25 | cjcld 15232 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (∗‘𝐵) ∈ ℂ) |
| 27 | 6, 5 | cphipcl 25239 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐶 , 𝐷) ∈ ℂ) |
| 28 | 27 | 3expb 1119 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , 𝐷) ∈ ℂ) |
| 29 | 28 | 3adant2 1130 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , 𝐷) ∈ ℂ) |
| 30 | 24, 26, 29 | mulassd 11282 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 · (∗‘𝐵)) · (𝐶 , 𝐷)) = (𝐴 · ((∗‘𝐵) · (𝐶 , 𝐷)))) |
| 31 | 12, 19, 30 | 3eqtr4d 2785 | 1 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 · 𝐶) , (𝐵 · 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 , 𝐷))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 · cmul 11158 ∗ccj 15132 Basecbs 17245 Scalarcsca 17301 ·𝑠 cvsca 17302 ·𝑖cip 17303 LModclmod 20875 ℂModcclm 25109 ℂPreHilccph 25214 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 ax-mulf 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-minusg 18968 df-subg 19154 df-ghm 19244 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-rhm 20489 df-subrg 20587 df-drng 20748 df-staf 20857 df-srng 20858 df-lmod 20877 df-lmhm 21039 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-cnfld 21383 df-phl 21662 df-nlm 24615 df-clm 25110 df-cph 25216 |
| This theorem is referenced by: pjthlem1 25485 |
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