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Mirrors > Home > MPE Home > Th. List > ip0l | Structured version Visualization version GIF version |
Description: Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (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 |
---|---|
ip0l | ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ( 0 , 𝐴) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phllmod 20319 | . . . . 5 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
2 | lmodgrp 19634 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
3 | phllmhm.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
4 | ip0l.o | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
5 | 3, 4 | grpidcl 18123 | . . . . 5 ⊢ (𝑊 ∈ Grp → 0 ∈ 𝑉) |
6 | 1, 2, 5 | 3syl 18 | . . . 4 ⊢ (𝑊 ∈ PreHil → 0 ∈ 𝑉) |
7 | 6 | adantr 484 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → 0 ∈ 𝑉) |
8 | oveq1 7142 | . . . 4 ⊢ (𝑥 = 0 → (𝑥 , 𝐴) = ( 0 , 𝐴)) | |
9 | eqid 2798 | . . . 4 ⊢ (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴)) = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴)) | |
10 | ovex 7168 | . . . 4 ⊢ ( 0 , 𝐴) ∈ V | |
11 | 8, 9, 10 | fvmpt 6745 | . . 3 ⊢ ( 0 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴))‘ 0 ) = ( 0 , 𝐴)) |
12 | 7, 11 | syl 17 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴))‘ 0 ) = ( 0 , 𝐴)) |
13 | phlsrng.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
14 | phllmhm.h | . . . 4 ⊢ , = (·𝑖‘𝑊) | |
15 | 13, 14, 3, 9 | phllmhm 20321 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴)) ∈ (𝑊 LMHom (ringLMod‘𝐹))) |
16 | lmghm 19796 | . . 3 ⊢ ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴)) ∈ (𝑊 GrpHom (ringLMod‘𝐹))) | |
17 | ip0l.z | . . . . 5 ⊢ 𝑍 = (0g‘𝐹) | |
18 | rlm0 19962 | . . . . 5 ⊢ (0g‘𝐹) = (0g‘(ringLMod‘𝐹)) | |
19 | 17, 18 | eqtri 2821 | . . . 4 ⊢ 𝑍 = (0g‘(ringLMod‘𝐹)) |
20 | 4, 19 | ghmid 18356 | . . 3 ⊢ ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴)) ∈ (𝑊 GrpHom (ringLMod‘𝐹)) → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴))‘ 0 ) = 𝑍) |
21 | 15, 16, 20 | 3syl 18 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴))‘ 0 ) = 𝑍) |
22 | 12, 21 | eqtr3d 2835 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ( 0 , 𝐴) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Scalarcsca 16560 ·𝑖cip 16562 0gc0g 16705 Grpcgrp 18095 GrpHom cghm 18347 LModclmod 19627 LMHom clmhm 19784 ringLModcrglmod 19934 PreHilcphl 20313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-sca 16573 df-vsca 16574 df-ip 16575 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-ghm 18348 df-lmod 19629 df-lmhm 19787 df-lvec 19868 df-sra 19937 df-rgmod 19938 df-phl 20315 |
This theorem is referenced by: ip0r 20326 ipeq0 20327 ocvlss 20361 cphip0l 23807 |
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