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Mirrors > Home > MPE Home > Th. List > frlmvscavalb | Structured version Visualization version GIF version |
Description: Scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.) |
Ref | Expression |
---|---|
frlmplusgvalb.f | ⊢ 𝐹 = (𝑅 freeLMod 𝐼) |
frlmplusgvalb.b | ⊢ 𝐵 = (Base‘𝐹) |
frlmplusgvalb.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
frlmplusgvalb.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
frlmplusgvalb.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
frlmplusgvalb.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
frlmvscavalb.k | ⊢ 𝐾 = (Base‘𝑅) |
frlmvscavalb.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
frlmvscavalb.v | ⊢ ∙ = ( ·𝑠 ‘𝐹) |
frlmvscavalb.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
frlmvscavalb | ⊢ (𝜑 → (𝑍 = (𝐴 ∙ 𝑋) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (𝐴 · (𝑋‘𝑖)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmplusgvalb.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
2 | frlmplusgvalb.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
3 | frlmplusgvalb.f | . . . . . . 7 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
4 | frlmvscavalb.k | . . . . . . 7 ⊢ 𝐾 = (Base‘𝑅) | |
5 | frlmplusgvalb.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐹) | |
6 | 3, 4, 5 | frlmbasmap 21306 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ (𝐾 ↑m 𝐼)) |
7 | 1, 2, 6 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝐾 ↑m 𝐼)) |
8 | 4 | fvexi 6903 | . . . . . . 7 ⊢ 𝐾 ∈ V |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ V) |
10 | 9, 1 | elmapd 8831 | . . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝐾 ↑m 𝐼) ↔ 𝑍:𝐼⟶𝐾)) |
11 | 7, 10 | mpbid 231 | . . . 4 ⊢ (𝜑 → 𝑍:𝐼⟶𝐾) |
12 | 11 | ffnd 6716 | . . 3 ⊢ (𝜑 → 𝑍 Fn 𝐼) |
13 | frlmplusgvalb.r | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
14 | 3 | frlmlmod 21296 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod) |
15 | 13, 1, 14 | syl2anc 585 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ LMod) |
16 | frlmvscavalb.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
17 | 16, 4 | eleqtrdi 2844 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅)) |
18 | 3 | frlmsca 21300 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘𝐹)) |
19 | 13, 1, 18 | syl2anc 585 | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹)) |
20 | 19 | fveq2d 6893 | . . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹))) |
21 | 17, 20 | eleqtrd 2836 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝐹))) |
22 | frlmplusgvalb.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
23 | eqid 2733 | . . . . . . . 8 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹) | |
24 | frlmvscavalb.v | . . . . . . . 8 ⊢ ∙ = ( ·𝑠 ‘𝐹) | |
25 | eqid 2733 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹)) | |
26 | 5, 23, 24, 25 | lmodvscl 20482 | . . . . . . 7 ⊢ ((𝐹 ∈ LMod ∧ 𝐴 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑋 ∈ 𝐵) → (𝐴 ∙ 𝑋) ∈ 𝐵) |
27 | 15, 21, 22, 26 | syl3anc 1372 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∙ 𝑋) ∈ 𝐵) |
28 | 3, 4, 5 | frlmbasmap 21306 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ (𝐴 ∙ 𝑋) ∈ 𝐵) → (𝐴 ∙ 𝑋) ∈ (𝐾 ↑m 𝐼)) |
29 | 1, 27, 28 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → (𝐴 ∙ 𝑋) ∈ (𝐾 ↑m 𝐼)) |
30 | 9, 1 | elmapd 8831 | . . . . 5 ⊢ (𝜑 → ((𝐴 ∙ 𝑋) ∈ (𝐾 ↑m 𝐼) ↔ (𝐴 ∙ 𝑋):𝐼⟶𝐾)) |
31 | 29, 30 | mpbid 231 | . . . 4 ⊢ (𝜑 → (𝐴 ∙ 𝑋):𝐼⟶𝐾) |
32 | 31 | ffnd 6716 | . . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) Fn 𝐼) |
33 | eqfnfv 7030 | . . 3 ⊢ ((𝑍 Fn 𝐼 ∧ (𝐴 ∙ 𝑋) Fn 𝐼) → (𝑍 = (𝐴 ∙ 𝑋) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 ∙ 𝑋)‘𝑖))) | |
34 | 12, 32, 33 | syl2anc 585 | . 2 ⊢ (𝜑 → (𝑍 = (𝐴 ∙ 𝑋) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 ∙ 𝑋)‘𝑖))) |
35 | 1 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ 𝑊) |
36 | 16 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐴 ∈ 𝐾) |
37 | 22 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑋 ∈ 𝐵) |
38 | simpr 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼) | |
39 | frlmvscavalb.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
40 | 3, 5, 4, 35, 36, 37, 38, 24, 39 | frlmvscaval 21315 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐴 ∙ 𝑋)‘𝑖) = (𝐴 · (𝑋‘𝑖))) |
41 | 40 | eqeq2d 2744 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑍‘𝑖) = ((𝐴 ∙ 𝑋)‘𝑖) ↔ (𝑍‘𝑖) = (𝐴 · (𝑋‘𝑖)))) |
42 | 41 | ralbidva 3176 | . 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 ∙ 𝑋)‘𝑖) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (𝐴 · (𝑋‘𝑖)))) |
43 | 34, 42 | bitrd 279 | 1 ⊢ (𝜑 → (𝑍 = (𝐴 ∙ 𝑋) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (𝐴 · (𝑋‘𝑖)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ↑m cmap 8817 Basecbs 17141 .rcmulr 17195 Scalarcsca 17197 ·𝑠 cvsca 17198 Ringcrg 20050 LModclmod 20464 freeLMod cfrlm 21293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-sup 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-hom 17218 df-cco 17219 df-0g 17384 df-prds 17390 df-pws 17392 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-grp 18819 df-minusg 18820 df-sbg 18821 df-subg 18998 df-mgp 19983 df-ur 20000 df-ring 20052 df-subrg 20354 df-lmod 20466 df-lss 20536 df-sra 20778 df-rgmod 20779 df-dsmm 21279 df-frlm 21294 |
This theorem is referenced by: (None) |
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