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Mirrors > Home > MPE Home > Th. List > frlmvplusgscavalb | Structured version Visualization version GIF version |
Description: Addition combined with 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) |
frlmvplusgscavalb.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
frlmvplusgscavalb.a | ⊢ + = (+g‘𝑅) |
frlmvplusgscavalb.p | ⊢ ✚ = (+g‘𝐹) |
frlmvplusgscavalb.c | ⊢ (𝜑 → 𝐶 ∈ 𝐾) |
Ref | Expression |
---|---|
frlmvplusgscavalb | ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmplusgvalb.f | . . 3 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
2 | frlmplusgvalb.b | . . 3 ⊢ 𝐵 = (Base‘𝐹) | |
3 | frlmplusgvalb.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
4 | frlmplusgvalb.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
5 | 1 | frlmlmod 21611 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod) |
6 | 4, 3, 5 | syl2anc 583 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ LMod) |
7 | frlmvscavalb.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
8 | frlmvscavalb.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑅) | |
9 | 7, 8 | eleqtrdi 2835 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅)) |
10 | 1 | frlmsca 21615 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘𝐹)) |
11 | 4, 3, 10 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹)) |
12 | 11 | fveq2d 6885 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹))) |
13 | 9, 12 | eleqtrd 2827 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝐹))) |
14 | frlmplusgvalb.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
15 | eqid 2724 | . . . . 5 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹) | |
16 | frlmvscavalb.v | . . . . 5 ⊢ ∙ = ( ·𝑠 ‘𝐹) | |
17 | eqid 2724 | . . . . 5 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹)) | |
18 | 2, 15, 16, 17 | lmodvscl 20713 | . . . 4 ⊢ ((𝐹 ∈ LMod ∧ 𝐴 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑋 ∈ 𝐵) → (𝐴 ∙ 𝑋) ∈ 𝐵) |
19 | 6, 13, 14, 18 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) ∈ 𝐵) |
20 | frlmplusgvalb.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
21 | frlmvplusgscavalb.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
22 | 21, 8 | eleqtrdi 2835 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝑅)) |
23 | 22, 12 | eleqtrd 2827 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (Base‘(Scalar‘𝐹))) |
24 | frlmvplusgscavalb.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
25 | 2, 15, 16, 17 | lmodvscl 20713 | . . . 4 ⊢ ((𝐹 ∈ LMod ∧ 𝐶 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑌 ∈ 𝐵) → (𝐶 ∙ 𝑌) ∈ 𝐵) |
26 | 6, 23, 24, 25 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝐶 ∙ 𝑌) ∈ 𝐵) |
27 | frlmvplusgscavalb.a | . . 3 ⊢ + = (+g‘𝑅) | |
28 | frlmvplusgscavalb.p | . . 3 ⊢ ✚ = (+g‘𝐹) | |
29 | 1, 2, 3, 19, 20, 4, 26, 27, 28 | frlmplusgvalb 21631 | . 2 ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)))) |
30 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ 𝑊) |
31 | 7 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐴 ∈ 𝐾) |
32 | 14 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑋 ∈ 𝐵) |
33 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼) | |
34 | frlmvscavalb.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
35 | 1, 2, 8, 30, 31, 32, 33, 16, 34 | frlmvscaval 21630 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐴 ∙ 𝑋)‘𝑖) = (𝐴 · (𝑋‘𝑖))) |
36 | 21 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐶 ∈ 𝐾) |
37 | 24 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑌 ∈ 𝐵) |
38 | 1, 2, 8, 30, 36, 37, 33, 16, 34 | frlmvscaval 21630 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐶 ∙ 𝑌)‘𝑖) = (𝐶 · (𝑌‘𝑖))) |
39 | 35, 38 | oveq12d 7419 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖)))) |
40 | 39 | eqeq2d 2735 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑍‘𝑖) = (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)) ↔ (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖))))) |
41 | 40 | ralbidva 3167 | . 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖))))) |
42 | 29, 41 | bitrd 279 | 1 ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ‘cfv 6533 (class class class)co 7401 Basecbs 17142 +gcplusg 17195 .rcmulr 17196 Scalarcsca 17198 ·𝑠 cvsca 17199 Ringcrg 20127 LModclmod 20695 freeLMod cfrlm 21608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-sup 9432 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-sca 17211 df-vsca 17212 df-ip 17213 df-tset 17214 df-ple 17215 df-ds 17217 df-hom 17219 df-cco 17220 df-0g 17385 df-prds 17391 df-pws 17393 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-grp 18855 df-minusg 18856 df-sbg 18857 df-subg 19039 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-subrg 20460 df-lmod 20697 df-lss 20768 df-sra 21010 df-rgmod 21011 df-dsmm 21594 df-frlm 21609 |
This theorem is referenced by: rrxplusgvscavalb 25244 |
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