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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 20821 | . . . . 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 2920 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅)) |
10 | 1 | frlmsca 20825 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘𝐹)) |
11 | 4, 3, 10 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹)) |
12 | 11 | fveq2d 6667 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹))) |
13 | 9, 12 | eleqtrd 2912 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝐹))) |
14 | frlmplusgvalb.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
15 | eqid 2818 | . . . . 5 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹) | |
16 | frlmvscavalb.v | . . . . 5 ⊢ ∙ = ( ·𝑠 ‘𝐹) | |
17 | eqid 2818 | . . . . 5 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹)) | |
18 | 2, 15, 16, 17 | lmodvscl 19580 | . . . 4 ⊢ ((𝐹 ∈ LMod ∧ 𝐴 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑋 ∈ 𝐵) → (𝐴 ∙ 𝑋) ∈ 𝐵) |
19 | 6, 13, 14, 18 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) ∈ 𝐵) |
20 | frlmplusgvalb.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
21 | frlmvplusgscavalb.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
22 | 21, 8 | eleqtrdi 2920 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝑅)) |
23 | 22, 12 | eleqtrd 2912 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (Base‘(Scalar‘𝐹))) |
24 | frlmvplusgscavalb.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
25 | 2, 15, 16, 17 | lmodvscl 19580 | . . . 4 ⊢ ((𝐹 ∈ LMod ∧ 𝐶 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑌 ∈ 𝐵) → (𝐶 ∙ 𝑌) ∈ 𝐵) |
26 | 6, 23, 24, 25 | syl3anc 1363 | . . 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 20841 | . 2 ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)))) |
30 | 3 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ 𝑊) |
31 | 7 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐴 ∈ 𝐾) |
32 | 14 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑋 ∈ 𝐵) |
33 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼) | |
34 | frlmvscavalb.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
35 | 1, 2, 8, 30, 31, 32, 33, 16, 34 | frlmvscaval 20840 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐴 ∙ 𝑋)‘𝑖) = (𝐴 · (𝑋‘𝑖))) |
36 | 21 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐶 ∈ 𝐾) |
37 | 24 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑌 ∈ 𝐵) |
38 | 1, 2, 8, 30, 36, 37, 33, 16, 34 | frlmvscaval 20840 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐶 ∙ 𝑌)‘𝑖) = (𝐶 · (𝑌‘𝑖))) |
39 | 35, 38 | oveq12d 7163 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖)))) |
40 | 39 | eqeq2d 2829 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑍‘𝑖) = (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)) ↔ (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖))))) |
41 | 40 | ralbidva 3193 | . 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 280 | 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 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 Scalarcsca 16556 ·𝑠 cvsca 16557 Ringcrg 19226 LModclmod 19563 freeLMod cfrlm 20818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-hom 16577 df-cco 16578 df-0g 16703 df-prds 16709 df-pws 16711 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-mgp 19169 df-ur 19181 df-ring 19228 df-subrg 19462 df-lmod 19565 df-lss 19633 df-sra 19873 df-rgmod 19874 df-dsmm 20804 df-frlm 20819 |
This theorem is referenced by: rrxplusgvscavalb 23925 |
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