|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 21769 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod) | 
| 6 | 4, 3, 5 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ LMod) | 
| 7 | frlmvscavalb.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
| 8 | frlmvscavalb.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑅) | |
| 9 | 7, 8 | eleqtrdi 2851 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅)) | 
| 10 | 1 | frlmsca 21773 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘𝐹)) | 
| 11 | 4, 3, 10 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹)) | 
| 12 | 11 | fveq2d 6910 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹))) | 
| 13 | 9, 12 | eleqtrd 2843 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝐹))) | 
| 14 | frlmplusgvalb.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 15 | eqid 2737 | . . . . 5 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹) | |
| 16 | frlmvscavalb.v | . . . . 5 ⊢ ∙ = ( ·𝑠 ‘𝐹) | |
| 17 | eqid 2737 | . . . . 5 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹)) | |
| 18 | 2, 15, 16, 17 | lmodvscl 20876 | . . . 4 ⊢ ((𝐹 ∈ LMod ∧ 𝐴 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑋 ∈ 𝐵) → (𝐴 ∙ 𝑋) ∈ 𝐵) | 
| 19 | 6, 13, 14, 18 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) ∈ 𝐵) | 
| 20 | frlmplusgvalb.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
| 21 | frlmvplusgscavalb.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
| 22 | 21, 8 | eleqtrdi 2851 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝑅)) | 
| 23 | 22, 12 | eleqtrd 2843 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (Base‘(Scalar‘𝐹))) | 
| 24 | frlmvplusgscavalb.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 25 | 2, 15, 16, 17 | lmodvscl 20876 | . . . 4 ⊢ ((𝐹 ∈ LMod ∧ 𝐶 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑌 ∈ 𝐵) → (𝐶 ∙ 𝑌) ∈ 𝐵) | 
| 26 | 6, 23, 24, 25 | syl3anc 1373 | . . 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 21789 | . 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 21788 | . . . . 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 21788 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐶 ∙ 𝑌)‘𝑖) = (𝐶 · (𝑌‘𝑖))) | 
| 39 | 35, 38 | oveq12d 7449 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖)))) | 
| 40 | 39 | eqeq2d 2748 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑍‘𝑖) = (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)) ↔ (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖))))) | 
| 41 | 40 | ralbidva 3176 | . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 Ringcrg 20230 LModclmod 20858 freeLMod cfrlm 21766 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-prds 17492 df-pws 17494 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-subrg 20570 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-dsmm 21752 df-frlm 21767 | 
| This theorem is referenced by: rrxplusgvscavalb 25429 | 
| Copyright terms: Public domain | W3C validator |