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Mirrors > Home > MPE Home > Th. List > ipsubdi | Structured version Visualization version GIF version |
Description: Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
phlsrng.f | ⊢ 𝐹 = (Scalar‘𝑊) |
phllmhm.h | ⊢ , = (·𝑖‘𝑊) |
phllmhm.v | ⊢ 𝑉 = (Base‘𝑊) |
ipsubdir.m | ⊢ − = (-g‘𝑊) |
ipsubdir.s | ⊢ 𝑆 = (-g‘𝐹) |
Ref | Expression |
---|---|
ipsubdi | ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , (𝐵 − 𝐶)) = ((𝐴 , 𝐵)𝑆(𝐴 , 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑊 ∈ PreHil) | |
2 | simpr1 1190 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝑉) | |
3 | phllmod 20774 | . . . . . . . 8 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
4 | 3 | adantr 483 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑊 ∈ LMod) |
5 | lmodgrp 19641 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑊 ∈ Grp) |
7 | simpr2 1191 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝑉) | |
8 | simpr3 1192 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
9 | phllmhm.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
10 | ipsubdir.m | . . . . . . 7 ⊢ − = (-g‘𝑊) | |
11 | 9, 10 | grpsubcl 18179 | . . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 − 𝐶) ∈ 𝑉) |
12 | 6, 7, 8, 11 | syl3anc 1367 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐵 − 𝐶) ∈ 𝑉) |
13 | phlsrng.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
14 | phllmhm.h | . . . . . 6 ⊢ , = (·𝑖‘𝑊) | |
15 | eqid 2821 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
16 | eqid 2821 | . . . . . 6 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
17 | 13, 14, 9, 15, 16 | ipdi 20784 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ (𝐵 − 𝐶) ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , ((𝐵 − 𝐶)(+g‘𝑊)𝐶)) = ((𝐴 , (𝐵 − 𝐶))(+g‘𝐹)(𝐴 , 𝐶))) |
18 | 1, 2, 12, 8, 17 | syl13anc 1368 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , ((𝐵 − 𝐶)(+g‘𝑊)𝐶)) = ((𝐴 , (𝐵 − 𝐶))(+g‘𝐹)(𝐴 , 𝐶))) |
19 | 9, 15, 10 | grpnpcan 18191 | . . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐵 − 𝐶)(+g‘𝑊)𝐶) = 𝐵) |
20 | 6, 7, 8, 19 | syl3anc 1367 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐵 − 𝐶)(+g‘𝑊)𝐶) = 𝐵) |
21 | 20 | oveq2d 7172 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , ((𝐵 − 𝐶)(+g‘𝑊)𝐶)) = (𝐴 , 𝐵)) |
22 | 18, 21 | eqtr3d 2858 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 , (𝐵 − 𝐶))(+g‘𝐹)(𝐴 , 𝐶)) = (𝐴 , 𝐵)) |
23 | 13 | lmodfgrp 19643 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp) |
24 | 4, 23 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐹 ∈ Grp) |
25 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
26 | 13, 14, 9, 25 | ipcl 20777 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ (Base‘𝐹)) |
27 | 1, 2, 7, 26 | syl3anc 1367 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , 𝐵) ∈ (Base‘𝐹)) |
28 | 13, 14, 9, 25 | ipcl 20777 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 , 𝐶) ∈ (Base‘𝐹)) |
29 | 1, 2, 8, 28 | syl3anc 1367 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , 𝐶) ∈ (Base‘𝐹)) |
30 | 13, 14, 9, 25 | ipcl 20777 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ (𝐵 − 𝐶) ∈ 𝑉) → (𝐴 , (𝐵 − 𝐶)) ∈ (Base‘𝐹)) |
31 | 1, 2, 12, 30 | syl3anc 1367 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , (𝐵 − 𝐶)) ∈ (Base‘𝐹)) |
32 | ipsubdir.s | . . . . 5 ⊢ 𝑆 = (-g‘𝐹) | |
33 | 25, 16, 32 | grpsubadd 18187 | . . . 4 ⊢ ((𝐹 ∈ Grp ∧ ((𝐴 , 𝐵) ∈ (Base‘𝐹) ∧ (𝐴 , 𝐶) ∈ (Base‘𝐹) ∧ (𝐴 , (𝐵 − 𝐶)) ∈ (Base‘𝐹))) → (((𝐴 , 𝐵)𝑆(𝐴 , 𝐶)) = (𝐴 , (𝐵 − 𝐶)) ↔ ((𝐴 , (𝐵 − 𝐶))(+g‘𝐹)(𝐴 , 𝐶)) = (𝐴 , 𝐵))) |
34 | 24, 27, 29, 31, 33 | syl13anc 1368 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (((𝐴 , 𝐵)𝑆(𝐴 , 𝐶)) = (𝐴 , (𝐵 − 𝐶)) ↔ ((𝐴 , (𝐵 − 𝐶))(+g‘𝐹)(𝐴 , 𝐶)) = (𝐴 , 𝐵))) |
35 | 22, 34 | mpbird 259 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 , 𝐵)𝑆(𝐴 , 𝐶)) = (𝐴 , (𝐵 − 𝐶))) |
36 | 35 | eqcomd 2827 | 1 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , (𝐵 − 𝐶)) = ((𝐴 , 𝐵)𝑆(𝐴 , 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 Scalarcsca 16568 ·𝑖cip 16570 Grpcgrp 18103 -gcsg 18105 LModclmod 19634 PreHilcphl 20768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-minusg 18107 df-sbg 18108 df-ghm 18356 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-rnghom 19467 df-staf 19616 df-srng 19617 df-lmod 19636 df-lmhm 19794 df-lvec 19875 df-sra 19944 df-rgmod 19945 df-phl 20770 |
This theorem is referenced by: ip2subdi 20788 ip2eq 20797 cphsubdi 23813 |
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