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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 21626 | . . . . 5 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
2 | lmodgrp 20843 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
3 | phllmhm.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
4 | ip0l.o | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
5 | 3, 4 | grpidcl 18960 | . . . . 5 ⊢ (𝑊 ∈ Grp → 0 ∈ 𝑉) |
6 | 1, 2, 5 | 3syl 18 | . . . 4 ⊢ (𝑊 ∈ PreHil → 0 ∈ 𝑉) |
7 | 6 | adantr 479 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → 0 ∈ 𝑉) |
8 | oveq1 7431 | . . . 4 ⊢ (𝑥 = 0 → (𝑥 , 𝐴) = ( 0 , 𝐴)) | |
9 | eqid 2726 | . . . 4 ⊢ (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴)) = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴)) | |
10 | ovex 7457 | . . . 4 ⊢ ( 0 , 𝐴) ∈ V | |
11 | 8, 9, 10 | fvmpt 7009 | . . 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 21628 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴)) ∈ (𝑊 LMHom (ringLMod‘𝐹))) |
16 | lmghm 21009 | . . 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 21181 | . . . . 5 ⊢ (0g‘𝐹) = (0g‘(ringLMod‘𝐹)) | |
19 | 17, 18 | eqtri 2754 | . . . 4 ⊢ 𝑍 = (0g‘(ringLMod‘𝐹)) |
20 | 4, 19 | ghmid 19216 | . . 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 2768 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ( 0 , 𝐴) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 Scalarcsca 17269 ·𝑖cip 17271 0gc0g 17454 Grpcgrp 18928 GrpHom cghm 19206 LModclmod 20836 LMHom clmhm 20997 ringLModcrglmod 21150 PreHilcphl 21620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-plusg 17279 df-sca 17282 df-vsca 17283 df-ip 17284 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-ghm 19207 df-lmod 20838 df-lmhm 21000 df-lvec 21081 df-sra 21151 df-rgmod 21152 df-phl 21622 |
This theorem is referenced by: ip0r 21633 ipeq0 21634 ocvlss 21668 cphip0l 25221 |
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