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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 20893 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod) |
6 | 4, 3, 5 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ LMod) |
7 | frlmvscavalb.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
8 | frlmvscavalb.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑅) | |
9 | 7, 8 | eleqtrdi 2923 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅)) |
10 | 1 | frlmsca 20897 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘𝐹)) |
11 | 4, 3, 10 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹)) |
12 | 11 | fveq2d 6674 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹))) |
13 | 9, 12 | eleqtrd 2915 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝐹))) |
14 | frlmplusgvalb.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
15 | eqid 2821 | . . . . 5 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹) | |
16 | frlmvscavalb.v | . . . . 5 ⊢ ∙ = ( ·𝑠 ‘𝐹) | |
17 | eqid 2821 | . . . . 5 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹)) | |
18 | 2, 15, 16, 17 | lmodvscl 19651 | . . . 4 ⊢ ((𝐹 ∈ LMod ∧ 𝐴 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑋 ∈ 𝐵) → (𝐴 ∙ 𝑋) ∈ 𝐵) |
19 | 6, 13, 14, 18 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) ∈ 𝐵) |
20 | frlmplusgvalb.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
21 | frlmvplusgscavalb.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
22 | 21, 8 | eleqtrdi 2923 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝑅)) |
23 | 22, 12 | eleqtrd 2915 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (Base‘(Scalar‘𝐹))) |
24 | frlmvplusgscavalb.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
25 | 2, 15, 16, 17 | lmodvscl 19651 | . . . 4 ⊢ ((𝐹 ∈ LMod ∧ 𝐶 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑌 ∈ 𝐵) → (𝐶 ∙ 𝑌) ∈ 𝐵) |
26 | 6, 23, 24, 25 | syl3anc 1367 | . . 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 20913 | . 2 ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)))) |
30 | 3 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ 𝑊) |
31 | 7 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐴 ∈ 𝐾) |
32 | 14 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑋 ∈ 𝐵) |
33 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼) | |
34 | frlmvscavalb.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
35 | 1, 2, 8, 30, 31, 32, 33, 16, 34 | frlmvscaval 20912 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐴 ∙ 𝑋)‘𝑖) = (𝐴 · (𝑋‘𝑖))) |
36 | 21 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐶 ∈ 𝐾) |
37 | 24 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑌 ∈ 𝐵) |
38 | 1, 2, 8, 30, 36, 37, 33, 16, 34 | frlmvscaval 20912 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐶 ∙ 𝑌)‘𝑖) = (𝐶 · (𝑌‘𝑖))) |
39 | 35, 38 | oveq12d 7174 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖)))) |
40 | 39 | eqeq2d 2832 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑍‘𝑖) = (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)) ↔ (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖))))) |
41 | 40 | ralbidva 3196 | . 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 281 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 .rcmulr 16566 Scalarcsca 16568 ·𝑠 cvsca 16569 Ringcrg 19297 LModclmod 19634 freeLMod cfrlm 20890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-hom 16589 df-cco 16590 df-0g 16715 df-prds 16721 df-pws 16723 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-mgp 19240 df-ur 19252 df-ring 19299 df-subrg 19533 df-lmod 19636 df-lss 19704 df-sra 19944 df-rgmod 19945 df-dsmm 20876 df-frlm 20891 |
This theorem is referenced by: rrxplusgvscavalb 23998 |
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