frlmvplusgscavalb - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > frlmvplusgscavalb		Structured version Visualization version GIF version

Theorem frlmvplusgscavalb 21633

Description: Addition combined with scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
frlmvplusgscavalb	⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖)))))

Distinct variable groups: 𝑖,𝐼 𝑖,𝑋 𝑖,𝑍 𝜑,𝑖 𝐴,𝑖 ∙ ,𝑖 𝐶,𝑖 𝑖,𝑌 ✚ ,𝑖 Allowed substitution hints: 𝐵(𝑖) + (𝑖) 𝑅(𝑖) · (𝑖) 𝐹(𝑖) 𝐾(𝑖) 𝑊(𝑖)

**Proof of Theorem frlmvplusgscavalb**
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

W3C validator