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Mirrors > Home > MPE Home > Th. List > ip0r | Structured version Visualization version GIF version |
Description: Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (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) |
ip0l.o | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
ip0r | ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → (𝐴 , 0 ) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phlsrng.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | phllmhm.h | . . . 4 ⊢ , = (·𝑖‘𝑊) | |
3 | phllmhm.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
4 | ip0l.z | . . . 4 ⊢ 𝑍 = (0g‘𝐹) | |
5 | ip0l.o | . . . 4 ⊢ 0 = (0g‘𝑊) | |
6 | 1, 2, 3, 4, 5 | ip0l 20497 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ( 0 , 𝐴) = 𝑍) |
7 | 6 | fveq2d 6500 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ((*𝑟‘𝐹)‘( 0 , 𝐴)) = ((*𝑟‘𝐹)‘𝑍)) |
8 | phllmod 20491 | . . . . 5 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
9 | 8 | adantr 473 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → 𝑊 ∈ LMod) |
10 | 3, 5 | lmod0vcl 19397 | . . . 4 ⊢ (𝑊 ∈ LMod → 0 ∈ 𝑉) |
11 | 9, 10 | syl 17 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → 0 ∈ 𝑉) |
12 | eqid 2771 | . . . . . 6 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹) | |
13 | 1, 2, 3, 12 | ipcj 20495 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 0 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ((*𝑟‘𝐹)‘( 0 , 𝐴)) = (𝐴 , 0 )) |
14 | 13 | 3expa 1099 | . . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 0 ∈ 𝑉) ∧ 𝐴 ∈ 𝑉) → ((*𝑟‘𝐹)‘( 0 , 𝐴)) = (𝐴 , 0 )) |
15 | 14 | an32s 640 | . . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) ∧ 0 ∈ 𝑉) → ((*𝑟‘𝐹)‘( 0 , 𝐴)) = (𝐴 , 0 )) |
16 | 11, 15 | mpdan 675 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ((*𝑟‘𝐹)‘( 0 , 𝐴)) = (𝐴 , 0 )) |
17 | 1 | phlsrng 20492 | . . . 4 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring) |
18 | 17 | adantr 473 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ *-Ring) |
19 | 12, 4 | srng0 19365 | . . 3 ⊢ (𝐹 ∈ *-Ring → ((*𝑟‘𝐹)‘𝑍) = 𝑍) |
20 | 18, 19 | syl 17 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ((*𝑟‘𝐹)‘𝑍) = 𝑍) |
21 | 7, 16, 20 | 3eqtr3d 2815 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → (𝐴 , 0 ) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ‘cfv 6185 (class class class)co 6974 Basecbs 16337 *𝑟cstv 16421 Scalarcsca 16422 ·𝑖cip 16424 0gc0g 16567 *-Ringcsr 19349 LModclmod 19368 PreHilcphl 20485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-tpos 7693 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-plusg 16432 df-mulr 16433 df-sca 16435 df-vsca 16436 df-ip 16437 df-0g 16569 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-mhm 17815 df-grp 17906 df-ghm 18139 df-mgp 18975 df-ur 18987 df-ring 19034 df-oppr 19108 df-rnghom 19202 df-staf 19350 df-srng 19351 df-lmod 19370 df-lmhm 19528 df-lvec 19609 df-sra 19678 df-rgmod 19679 df-phl 20487 |
This theorem is referenced by: cphip0r 23525 ipcau2 23555 |
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