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Mirrors > Home > MPE Home > Th. List > Mathboxes > lfladd | Structured version Visualization version GIF version |
Description: Property of a linear functional. (lnfnaddi 31866 analog.) (Contributed by NM, 18-Apr-2014.) |
Ref | Expression |
---|---|
lfladd.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lfladd.p | ⊢ ⨣ = (+g‘𝐷) |
lfladd.v | ⊢ 𝑉 = (Base‘𝑊) |
lfladd.a | ⊢ + = (+g‘𝑊) |
lfladd.f | ⊢ 𝐹 = (LFnl‘𝑊) |
Ref | Expression |
---|---|
lfladd | ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐺‘(𝑋 + 𝑌)) = ((𝐺‘𝑋) ⨣ (𝐺‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1134 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → 𝑊 ∈ LMod) | |
2 | simp2 1135 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → 𝐺 ∈ 𝐹) | |
3 | lfladd.d | . . . . 5 ⊢ 𝐷 = (Scalar‘𝑊) | |
4 | eqid 2728 | . . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
5 | eqid 2728 | . . . . 5 ⊢ (1r‘𝐷) = (1r‘𝐷) | |
6 | 3, 4, 5 | lmod1cl 20772 | . . . 4 ⊢ (𝑊 ∈ LMod → (1r‘𝐷) ∈ (Base‘𝐷)) |
7 | 6 | 3ad2ant1 1131 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (1r‘𝐷) ∈ (Base‘𝐷)) |
8 | simp3l 1199 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → 𝑋 ∈ 𝑉) | |
9 | simp3r 1200 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → 𝑌 ∈ 𝑉) | |
10 | lfladd.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
11 | lfladd.a | . . . 4 ⊢ + = (+g‘𝑊) | |
12 | eqid 2728 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
13 | lfladd.p | . . . 4 ⊢ ⨣ = (+g‘𝐷) | |
14 | eqid 2728 | . . . 4 ⊢ (.r‘𝐷) = (.r‘𝐷) | |
15 | lfladd.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
16 | 10, 11, 3, 12, 4, 13, 14, 15 | lfli 38533 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ((1r‘𝐷) ∈ (Base‘𝐷) ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐺‘(((1r‘𝐷)( ·𝑠 ‘𝑊)𝑋) + 𝑌)) = (((1r‘𝐷)(.r‘𝐷)(𝐺‘𝑋)) ⨣ (𝐺‘𝑌))) |
17 | 1, 2, 7, 8, 9, 16 | syl113anc 1380 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐺‘(((1r‘𝐷)( ·𝑠 ‘𝑊)𝑋) + 𝑌)) = (((1r‘𝐷)(.r‘𝐷)(𝐺‘𝑋)) ⨣ (𝐺‘𝑌))) |
18 | 10, 3, 12, 5 | lmodvs1 20773 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((1r‘𝐷)( ·𝑠 ‘𝑊)𝑋) = 𝑋) |
19 | 1, 8, 18 | syl2anc 583 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((1r‘𝐷)( ·𝑠 ‘𝑊)𝑋) = 𝑋) |
20 | 19 | fvoveq1d 7442 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐺‘(((1r‘𝐷)( ·𝑠 ‘𝑊)𝑋) + 𝑌)) = (𝐺‘(𝑋 + 𝑌))) |
21 | 3 | lmodring 20751 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝐷 ∈ Ring) |
22 | 21 | 3ad2ant1 1131 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → 𝐷 ∈ Ring) |
23 | 3, 4, 10, 15 | lflcl 38536 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ (Base‘𝐷)) |
24 | 23 | 3adant3r 1179 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐺‘𝑋) ∈ (Base‘𝐷)) |
25 | 4, 14, 5 | ringlidm 20205 | . . . 4 ⊢ ((𝐷 ∈ Ring ∧ (𝐺‘𝑋) ∈ (Base‘𝐷)) → ((1r‘𝐷)(.r‘𝐷)(𝐺‘𝑋)) = (𝐺‘𝑋)) |
26 | 22, 24, 25 | syl2anc 583 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((1r‘𝐷)(.r‘𝐷)(𝐺‘𝑋)) = (𝐺‘𝑋)) |
27 | 26 | oveq1d 7435 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((1r‘𝐷)(.r‘𝐷)(𝐺‘𝑋)) ⨣ (𝐺‘𝑌)) = ((𝐺‘𝑋) ⨣ (𝐺‘𝑌))) |
28 | 17, 20, 27 | 3eqtr3d 2776 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐺‘(𝑋 + 𝑌)) = ((𝐺‘𝑋) ⨣ (𝐺‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 +gcplusg 17233 .rcmulr 17234 Scalarcsca 17236 ·𝑠 cvsca 17237 1rcur 20121 Ringcrg 20173 LModclmod 20743 LFnlclfn 38529 |
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 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mgp 20075 df-ur 20122 df-ring 20175 df-lmod 20745 df-lfl 38530 |
This theorem is referenced by: lfladdcl 38543 hdmaplna1 41380 |
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