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Mirrors > Home > MPE Home > Th. List > lmodsubvs | Structured version Visualization version GIF version |
Description: Subtraction of a scalar product in terms of addition. (Contributed by NM, 9-Apr-2015.) |
Ref | Expression |
---|---|
lmodsubvs.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodsubvs.p | ⊢ + = (+g‘𝑊) |
lmodsubvs.m | ⊢ − = (-g‘𝑊) |
lmodsubvs.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lmodsubvs.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lmodsubvs.k | ⊢ 𝐾 = (Base‘𝐹) |
lmodsubvs.n | ⊢ 𝑁 = (invg‘𝐹) |
lmodsubvs.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lmodsubvs.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
lmodsubvs.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lmodsubvs.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
lmodsubvs | ⊢ (𝜑 → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘𝐴) · 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodsubvs.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lmodsubvs.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | lmodsubvs.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
4 | lmodsubvs.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
5 | lmodsubvs.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
6 | lmodsubvs.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
7 | lmodsubvs.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
8 | lmodsubvs.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
9 | 5, 6, 7, 8 | lmodvscl 19644 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝐴 · 𝑌) ∈ 𝑉) |
10 | 1, 3, 4, 9 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝐴 · 𝑌) ∈ 𝑉) |
11 | lmodsubvs.p | . . . 4 ⊢ + = (+g‘𝑊) | |
12 | lmodsubvs.m | . . . 4 ⊢ − = (-g‘𝑊) | |
13 | lmodsubvs.n | . . . 4 ⊢ 𝑁 = (invg‘𝐹) | |
14 | eqid 2798 | . . . 4 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
15 | 5, 11, 12, 6, 7, 13, 14 | lmodvsubval2 19682 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ (𝐴 · 𝑌) ∈ 𝑉) → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌)))) |
16 | 1, 2, 10, 15 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌)))) |
17 | 6 | lmodring 19635 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
18 | 1, 17 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ Ring) |
19 | ringgrp 19295 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
20 | 18, 19 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ Grp) |
21 | 8, 14 | ringidcl 19314 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ 𝐾) |
22 | 18, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → (1r‘𝐹) ∈ 𝐾) |
23 | 8, 13 | grpinvcl 18143 | . . . . . 6 ⊢ ((𝐹 ∈ Grp ∧ (1r‘𝐹) ∈ 𝐾) → (𝑁‘(1r‘𝐹)) ∈ 𝐾) |
24 | 20, 22, 23 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (𝑁‘(1r‘𝐹)) ∈ 𝐾) |
25 | eqid 2798 | . . . . . 6 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
26 | 5, 6, 7, 8, 25 | lmodvsass 19652 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ ((𝑁‘(1r‘𝐹)) ∈ 𝐾 ∧ 𝐴 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉)) → (((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) · 𝑌) = ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌))) |
27 | 1, 24, 3, 4, 26 | syl13anc 1369 | . . . 4 ⊢ (𝜑 → (((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) · 𝑌) = ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌))) |
28 | 8, 25, 14, 13, 18, 3 | ringnegl 19340 | . . . . 5 ⊢ (𝜑 → ((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) = (𝑁‘𝐴)) |
29 | 28 | oveq1d 7150 | . . . 4 ⊢ (𝜑 → (((𝑁‘(1r‘𝐹))(.r‘𝐹)𝐴) · 𝑌) = ((𝑁‘𝐴) · 𝑌)) |
30 | 27, 29 | eqtr3d 2835 | . . 3 ⊢ (𝜑 → ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌)) = ((𝑁‘𝐴) · 𝑌)) |
31 | 30 | oveq2d 7151 | . 2 ⊢ (𝜑 → (𝑋 + ((𝑁‘(1r‘𝐹)) · (𝐴 · 𝑌))) = (𝑋 + ((𝑁‘𝐴) · 𝑌))) |
32 | 16, 31 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘𝐴) · 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 Scalarcsca 16560 ·𝑠 cvsca 16561 Grpcgrp 18095 invgcminusg 18096 -gcsg 18097 1rcur 19244 Ringcrg 19290 LModclmod 19627 |
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-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-1st 7671 df-2nd 7672 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-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mgp 19233 df-ur 19245 df-ring 19292 df-lmod 19629 |
This theorem is referenced by: lspexch 19894 baerlem5alem1 39004 baerlem5blem1 39005 |
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