Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 20166 | . . . 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 20136 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐴 · 𝑋) ∈ 𝑉) |
11 | 1, 4, 5, 10 | syl3anc 1370 | . . 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 20136 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝐵 · 𝑌) ∈ 𝑉) |
15 | 1, 12, 13, 14 | syl3anc 1370 | . . 3 ⊢ (𝜑 → (𝐵 · 𝑌) ∈ 𝑉) |
16 | lmodnegadd.p | . . . 4 ⊢ + = (+g‘𝑊) | |
17 | lmodnegadd.n | . . . 4 ⊢ 𝑁 = (invg‘𝑊) | |
18 | 6, 16, 17 | ablinvadd 19407 | . . 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 1370 | . 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 20163 | . . 3 ⊢ (𝜑 → (𝑁‘(𝐴 · 𝑋)) = ((𝐼‘𝐴) · 𝑋)) |
22 | 6, 7, 8, 17, 9, 20, 1, 13, 12 | lmodvsneg 20163 | . . 3 ⊢ (𝜑 → (𝑁‘(𝐵 · 𝑌)) = ((𝐼‘𝐵) · 𝑌)) |
23 | 21, 22 | oveq12d 7287 | . 2 ⊢ (𝜑 → ((𝑁‘(𝐴 · 𝑋)) + (𝑁‘(𝐵 · 𝑌))) = (((𝐼‘𝐴) · 𝑋) + ((𝐼‘𝐵) · 𝑌))) |
24 | 19, 23 | eqtrd 2780 | 1 ⊢ (𝜑 → (𝑁‘((𝐴 · 𝑋) + (𝐵 · 𝑌))) = (((𝐼‘𝐴) · 𝑋) + ((𝐼‘𝐵) · 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ‘cfv 6431 (class class class)co 7269 Basecbs 16908 +gcplusg 16958 Scalarcsca 16961 ·𝑠 cvsca 16962 invgcminusg 18574 Abelcabl 19383 LModclmod 20119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8479 df-en 8715 df-dom 8716 df-sdom 8717 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-plusg 16971 df-0g 17148 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-grp 18576 df-minusg 18577 df-cmn 19384 df-abl 19385 df-mgp 19717 df-ur 19734 df-ring 19781 df-lmod 20121 |
This theorem is referenced by: baerlem3lem1 39715 |
Copyright terms: Public domain | W3C validator |