Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > iporthcom | Structured version Visualization version GIF version |
Description: Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
phlsrng.f | ⊢ 𝐹 = (Scalar‘𝑊) |
phllmhm.h | ⊢ , = (·𝑖‘𝑊) |
phllmhm.v | ⊢ 𝑉 = (Base‘𝑊) |
ip0l.z | ⊢ 𝑍 = (0g‘𝐹) |
Ref | Expression |
---|---|
iporthcom | ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 , 𝐵) = 𝑍 ↔ (𝐵 , 𝐴) = 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phlsrng.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | 1 | phlsrng 19795 | . . . . 5 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring) |
3 | 2 | 3ad2ant1 1075 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐹 ∈ *-Ring) |
4 | eqid 2610 | . . . . 5 ⊢ (*rf‘𝐹) = (*rf‘𝐹) | |
5 | eqid 2610 | . . . . 5 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
6 | 4, 5 | srngf1o 18677 | . . . 4 ⊢ (𝐹 ∈ *-Ring → (*rf‘𝐹):(Base‘𝐹)–1-1-onto→(Base‘𝐹)) |
7 | f1of1 6049 | . . . 4 ⊢ ((*rf‘𝐹):(Base‘𝐹)–1-1-onto→(Base‘𝐹) → (*rf‘𝐹):(Base‘𝐹)–1-1→(Base‘𝐹)) | |
8 | 3, 6, 7 | 3syl 18 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (*rf‘𝐹):(Base‘𝐹)–1-1→(Base‘𝐹)) |
9 | phllmhm.h | . . . 4 ⊢ , = (·𝑖‘𝑊) | |
10 | phllmhm.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
11 | 1, 9, 10, 5 | ipcl 19797 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ (Base‘𝐹)) |
12 | phllmod 19794 | . . . . 5 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
13 | 12 | 3ad2ant1 1075 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ LMod) |
14 | ip0l.z | . . . . 5 ⊢ 𝑍 = (0g‘𝐹) | |
15 | 1, 5, 14 | lmod0cl 18712 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑍 ∈ (Base‘𝐹)) |
16 | 13, 15 | syl 17 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑍 ∈ (Base‘𝐹)) |
17 | f1fveq 6420 | . . 3 ⊢ (((*rf‘𝐹):(Base‘𝐹)–1-1→(Base‘𝐹) ∧ ((𝐴 , 𝐵) ∈ (Base‘𝐹) ∧ 𝑍 ∈ (Base‘𝐹))) → (((*rf‘𝐹)‘(𝐴 , 𝐵)) = ((*rf‘𝐹)‘𝑍) ↔ (𝐴 , 𝐵) = 𝑍)) | |
18 | 8, 11, 16, 17 | syl12anc 1316 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (((*rf‘𝐹)‘(𝐴 , 𝐵)) = ((*rf‘𝐹)‘𝑍) ↔ (𝐴 , 𝐵) = 𝑍)) |
19 | eqid 2610 | . . . . . 6 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹) | |
20 | 5, 19, 4 | stafval 18671 | . . . . 5 ⊢ ((𝐴 , 𝐵) ∈ (Base‘𝐹) → ((*rf‘𝐹)‘(𝐴 , 𝐵)) = ((*𝑟‘𝐹)‘(𝐴 , 𝐵))) |
21 | 11, 20 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((*rf‘𝐹)‘(𝐴 , 𝐵)) = ((*𝑟‘𝐹)‘(𝐴 , 𝐵))) |
22 | 1, 9, 10, 19 | ipcj 19798 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((*𝑟‘𝐹)‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)) |
23 | 21, 22 | eqtrd 2644 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((*rf‘𝐹)‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)) |
24 | 5, 19, 4 | stafval 18671 | . . . . 5 ⊢ (𝑍 ∈ (Base‘𝐹) → ((*rf‘𝐹)‘𝑍) = ((*𝑟‘𝐹)‘𝑍)) |
25 | 16, 24 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((*rf‘𝐹)‘𝑍) = ((*𝑟‘𝐹)‘𝑍)) |
26 | 19, 14 | srng0 18683 | . . . . 5 ⊢ (𝐹 ∈ *-Ring → ((*𝑟‘𝐹)‘𝑍) = 𝑍) |
27 | 3, 26 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((*𝑟‘𝐹)‘𝑍) = 𝑍) |
28 | 25, 27 | eqtrd 2644 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((*rf‘𝐹)‘𝑍) = 𝑍) |
29 | 23, 28 | eqeq12d 2625 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (((*rf‘𝐹)‘(𝐴 , 𝐵)) = ((*rf‘𝐹)‘𝑍) ↔ (𝐵 , 𝐴) = 𝑍)) |
30 | 18, 29 | bitr3d 269 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 , 𝐵) = 𝑍 ↔ (𝐵 , 𝐴) = 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 –1-1→wf1 5801 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 *𝑟cstv 15770 Scalarcsca 15771 ·𝑖cip 15773 0gc0g 15923 *rfcstf 18666 *-Ringcsr 18667 LModclmod 18686 PreHilcphl 19788 |
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 |
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-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-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 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-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-ghm 17481 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-rnghom 18538 df-staf 18668 df-srng 18669 df-lmod 18688 df-lmhm 18843 df-lvec 18924 df-sra 18993 df-rgmod 18994 df-phl 19790 |
This theorem is referenced by: ocvocv 19834 lsmcss 19855 cphorthcom 22809 |
Copyright terms: Public domain | W3C validator |