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Mirrors > Home > MPE Home > Th. List > lmodnegadd | Structured version Visualization version GIF version |
Description: Distribute negation through addition of scalar products. (Contributed by NM, 9-Apr-2015.) |
Ref | Expression |
---|---|
lmodnegadd.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodnegadd.p | ⊢ + = (+g‘𝑊) |
lmodnegadd.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lmodnegadd.n | ⊢ 𝑁 = (invg‘𝑊) |
lmodnegadd.r | ⊢ 𝑅 = (Scalar‘𝑊) |
lmodnegadd.k | ⊢ 𝐾 = (Base‘𝑅) |
lmodnegadd.i | ⊢ 𝐼 = (invg‘𝑅) |
lmodnegadd.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lmodnegadd.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
lmodnegadd.b | ⊢ (𝜑 → 𝐵 ∈ 𝐾) |
lmodnegadd.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lmodnegadd.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
lmodnegadd | ⊢ (𝜑 → (𝑁‘((𝐴 · 𝑋) + (𝐵 · 𝑌))) = (((𝐼‘𝐴) · 𝑋) + ((𝐼‘𝐵) · 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodnegadd.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lmodabl 19674 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝑊 ∈ Abel) |
4 | lmodnegadd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
5 | lmodnegadd.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
6 | lmodnegadd.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
7 | lmodnegadd.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑊) | |
8 | lmodnegadd.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
9 | lmodnegadd.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
10 | 6, 7, 8, 9 | lmodvscl 19644 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐴 · 𝑋) ∈ 𝑉) |
11 | 1, 4, 5, 10 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ 𝑉) |
12 | lmodnegadd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
13 | lmodnegadd.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
14 | 6, 7, 8, 9 | lmodvscl 19644 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝐵 · 𝑌) ∈ 𝑉) |
15 | 1, 12, 13, 14 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝐵 · 𝑌) ∈ 𝑉) |
16 | lmodnegadd.p | . . . 4 ⊢ + = (+g‘𝑊) | |
17 | lmodnegadd.n | . . . 4 ⊢ 𝑁 = (invg‘𝑊) | |
18 | 6, 16, 17 | ablinvadd 18923 | . . 3 ⊢ ((𝑊 ∈ Abel ∧ (𝐴 · 𝑋) ∈ 𝑉 ∧ (𝐵 · 𝑌) ∈ 𝑉) → (𝑁‘((𝐴 · 𝑋) + (𝐵 · 𝑌))) = ((𝑁‘(𝐴 · 𝑋)) + (𝑁‘(𝐵 · 𝑌)))) |
19 | 3, 11, 15, 18 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝑁‘((𝐴 · 𝑋) + (𝐵 · 𝑌))) = ((𝑁‘(𝐴 · 𝑋)) + (𝑁‘(𝐵 · 𝑌)))) |
20 | lmodnegadd.i | . . . 4 ⊢ 𝐼 = (invg‘𝑅) | |
21 | 6, 7, 8, 17, 9, 20, 1, 5, 4 | lmodvsneg 19671 | . . 3 ⊢ (𝜑 → (𝑁‘(𝐴 · 𝑋)) = ((𝐼‘𝐴) · 𝑋)) |
22 | 6, 7, 8, 17, 9, 20, 1, 13, 12 | lmodvsneg 19671 | . . 3 ⊢ (𝜑 → (𝑁‘(𝐵 · 𝑌)) = ((𝐼‘𝐵) · 𝑌)) |
23 | 21, 22 | oveq12d 7153 | . 2 ⊢ (𝜑 → ((𝑁‘(𝐴 · 𝑋)) + (𝑁‘(𝐵 · 𝑌))) = (((𝐼‘𝐴) · 𝑋) + ((𝐼‘𝐵) · 𝑌))) |
24 | 19, 23 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝑁‘((𝐴 · 𝑋) + (𝐵 · 𝑌))) = (((𝐼‘𝐴) · 𝑋) + ((𝐼‘𝐵) · 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 Scalarcsca 16560 ·𝑠 cvsca 16561 invgcminusg 18096 Abelcabl 18899 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-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-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-lmod 19629 |
This theorem is referenced by: baerlem3lem1 39003 |
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