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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 21619 | . . . . 5 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring) |
3 | 2 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐹 ∈ *-Ring) |
4 | eqid 2725 | . . . . 5 ⊢ (*rf‘𝐹) = (*rf‘𝐹) | |
5 | eqid 2725 | . . . . 5 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
6 | 4, 5 | srngf1o 20774 | . . . 4 ⊢ (𝐹 ∈ *-Ring → (*rf‘𝐹):(Base‘𝐹)–1-1-onto→(Base‘𝐹)) |
7 | f1of1 6841 | . . . 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 21621 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ (Base‘𝐹)) |
12 | phllmod 21618 | . . . . 5 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
13 | 12 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ LMod) |
14 | ip0l.z | . . . . 5 ⊢ 𝑍 = (0g‘𝐹) | |
15 | 1, 5, 14 | lmod0cl 20811 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑍 ∈ (Base‘𝐹)) |
16 | 13, 15 | syl 17 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑍 ∈ (Base‘𝐹)) |
17 | f1fveq 7276 | . . 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 835 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (((*rf‘𝐹)‘(𝐴 , 𝐵)) = ((*rf‘𝐹)‘𝑍) ↔ (𝐴 , 𝐵) = 𝑍)) |
19 | eqid 2725 | . . . . . 6 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹) | |
20 | 5, 19, 4 | stafval 20768 | . . . . 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 21622 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((*𝑟‘𝐹)‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)) |
23 | 21, 22 | eqtrd 2765 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((*rf‘𝐹)‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)) |
24 | 5, 19, 4 | stafval 20768 | . . . . 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 20780 | . . . . 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 2765 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((*rf‘𝐹)‘𝑍) = 𝑍) |
29 | 23, 28 | eqeq12d 2741 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (((*rf‘𝐹)‘(𝐴 , 𝐵)) = ((*rf‘𝐹)‘𝑍) ↔ (𝐵 , 𝐴) = 𝑍)) |
30 | 18, 29 | bitr3d 280 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 , 𝐵) = 𝑍 ↔ (𝐵 , 𝐴) = 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 –1-1→wf1 6550 –1-1-onto→wf1o 6552 ‘cfv 6553 (class class class)co 7423 Basecbs 17208 *𝑟cstv 17263 Scalarcsca 17264 ·𝑖cip 17266 0gc0g 17449 *rfcstf 20763 *-Ringcsr 20764 LModclmod 20783 PreHilcphl 21612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-map 8856 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-plusg 17274 df-mulr 17275 df-sca 17277 df-vsca 17278 df-ip 17279 df-0g 17451 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-grp 18926 df-ghm 19202 df-mgp 20113 df-ur 20160 df-ring 20213 df-oppr 20311 df-rhm 20449 df-staf 20765 df-srng 20766 df-lmod 20785 df-lmhm 20947 df-lvec 21028 df-sra 21098 df-rgmod 21099 df-phl 21614 |
This theorem is referenced by: ocvocv 21659 lsmcss 21680 cphorthcom 25212 |
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