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Mirrors > Home > MPE Home > Th. List > ipassr2 | Structured version Visualization version GIF version |
Description: "Associative" law for inner product. Conjugate version of ipassr 20793. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
phlsrng.f | ⊢ 𝐹 = (Scalar‘𝑊) |
phllmhm.h | ⊢ , = (·𝑖‘𝑊) |
phllmhm.v | ⊢ 𝑉 = (Base‘𝑊) |
ipdir.f | ⊢ 𝐾 = (Base‘𝐹) |
ipass.s | ⊢ · = ( ·𝑠 ‘𝑊) |
ipass.p | ⊢ × = (.r‘𝐹) |
ipassr.i | ⊢ ∗ = (*𝑟‘𝐹) |
Ref | Expression |
---|---|
ipassr2 | ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ((𝐴 , 𝐵) × 𝐶) = (𝐴 , (( ∗ ‘𝐶) · 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝑊 ∈ PreHil) | |
2 | simpr1 1190 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐴 ∈ 𝑉) | |
3 | simpr2 1191 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐵 ∈ 𝑉) | |
4 | phlsrng.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
5 | 4 | phlsrng 20778 | . . . 4 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring) |
6 | simpr3 1192 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐶 ∈ 𝐾) | |
7 | ipassr.i | . . . . 5 ⊢ ∗ = (*𝑟‘𝐹) | |
8 | ipdir.f | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
9 | 7, 8 | srngcl 19629 | . . . 4 ⊢ ((𝐹 ∈ *-Ring ∧ 𝐶 ∈ 𝐾) → ( ∗ ‘𝐶) ∈ 𝐾) |
10 | 5, 6, 9 | syl2an2r 683 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘𝐶) ∈ 𝐾) |
11 | phllmhm.h | . . . 4 ⊢ , = (·𝑖‘𝑊) | |
12 | phllmhm.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
13 | ipass.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
14 | ipass.p | . . . 4 ⊢ × = (.r‘𝐹) | |
15 | 4, 11, 12, 8, 13, 14, 7 | ipassr 20793 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ ( ∗ ‘𝐶) ∈ 𝐾)) → (𝐴 , (( ∗ ‘𝐶) · 𝐵)) = ((𝐴 , 𝐵) × ( ∗ ‘( ∗ ‘𝐶)))) |
16 | 1, 2, 3, 10, 15 | syl13anc 1368 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐴 , (( ∗ ‘𝐶) · 𝐵)) = ((𝐴 , 𝐵) × ( ∗ ‘( ∗ ‘𝐶)))) |
17 | 7, 8 | srngnvl 19630 | . . . 4 ⊢ ((𝐹 ∈ *-Ring ∧ 𝐶 ∈ 𝐾) → ( ∗ ‘( ∗ ‘𝐶)) = 𝐶) |
18 | 5, 6, 17 | syl2an2r 683 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘( ∗ ‘𝐶)) = 𝐶) |
19 | 18 | oveq2d 7175 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ((𝐴 , 𝐵) × ( ∗ ‘( ∗ ‘𝐶))) = ((𝐴 , 𝐵) × 𝐶)) |
20 | 16, 19 | eqtr2d 2860 | 1 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ((𝐴 , 𝐵) × 𝐶) = (𝐴 , (( ∗ ‘𝐶) · 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 .rcmulr 16569 *𝑟cstv 16570 Scalarcsca 16571 ·𝑠 cvsca 16572 ·𝑖cip 16573 *-Ringcsr 19618 PreHilcphl 20771 |
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 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-ip 16586 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-ghm 18359 df-mgp 19243 df-ur 19255 df-ring 19302 df-oppr 19376 df-rnghom 19470 df-staf 19619 df-srng 19620 df-lmod 19639 df-lmhm 19797 df-lvec 19878 df-sra 19947 df-rgmod 19948 df-phl 20773 |
This theorem is referenced by: ipcau2 23840 |
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