frlmvplusgscavalb - Metamath Proof Explorer

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Theorem frlmvplusgscavalb 21973

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 21951	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod)
6	4, 3, 5	syl2anc 596	. . . 4 ⊢ (𝜑 → 𝐹 ∈ LMod)
7		frlmvscavalb.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐾)
8		frlmvscavalb.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑅)
9	7, 8	eleqtrdi 2875	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅))
10	1	frlmsca 21955	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘𝐹))
11	4, 3, 10	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹))
12	11	fveq2d 6889	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
13	9, 12	eleqtrd 2867	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝐹)))
14		frlmplusgvalb.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
15		eqid 2765	. . . . 5 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹)
16		frlmvscavalb.v	. . . . 5 ⊢ ∙ = ( ·_𝑠 ‘𝐹)
17		eqid 2765	. . . . 5 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
18	2, 15, 16, 17	lmodvscl 21051	. . . 4 ⊢ ((𝐹 ∈ LMod ∧ 𝐴 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑋 ∈ 𝐵) → (𝐴 ∙ 𝑋) ∈ 𝐵)
19	6, 13, 14, 18	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) ∈ 𝐵)
20		frlmplusgvalb.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
21		frlmvplusgscavalb.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
22	21, 8	eleqtrdi 2875	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝑅))
23	22, 12	eleqtrd 2867	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (Base‘(Scalar‘𝐹)))
24		frlmvplusgscavalb.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
25	2, 15, 16, 17	lmodvscl 21051	. . . 4 ⊢ ((𝐹 ∈ LMod ∧ 𝐶 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑌 ∈ 𝐵) → (𝐶 ∙ 𝑌) ∈ 𝐵)
26	6, 23, 24, 25	syl3anc 1398	. . 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 21971	. 2 ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖))))
30	3	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ 𝑊)
31	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐴 ∈ 𝐾)
32	14	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑋 ∈ 𝐵)
33		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
34		frlmvscavalb.t	. . . . . 6 ⊢ · = (._r‘𝑅)
35	1, 2, 8, 30, 31, 32, 33, 16, 34	frlmvscaval 21970	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐴 ∙ 𝑋)‘𝑖) = (𝐴 · (𝑋‘𝑖)))
36	21	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐶 ∈ 𝐾)
37	24	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑌 ∈ 𝐵)
38	1, 2, 8, 30, 36, 37, 33, 16, 34	frlmvscaval 21970	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐶 ∙ 𝑌)‘𝑖) = (𝐶 · (𝑌‘𝑖)))
39	35, 38	oveq12d 7437	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖))))
40	39	eqeq2d 2776	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑍‘𝑖) = (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)) ↔ (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖)))))
41	40	ralbidva 3188	. 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 282	1 ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖)))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∀wral 3081 ‘cfv 6540 (class class class)co 7419 Basecbs 17293 +_gcplusg 17334 ._rcmulr 17335 Scalarcsca 17337 ·_𝑠 cvsca 17338 Ringcrg 20361 LModclmod 21033 freeLMod cfrlm 21948

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-sup 9409 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12522 df-z 12609 df-dec 12730 df-uz 12881 df-fz 13554 df-struct 17231 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-mulr 17348 df-sca 17350 df-vsca 17351 df-ip 17352 df-tset 17353 df-ple 17354 df-ds 17356 df-hom 17358 df-cco 17359 df-0g 17518 df-prds 17524 df-pws 17526 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-grp 19049 df-minusg 19050 df-sbg 19051 df-subg 19235 df-cmn 19898 df-abl 19899 df-mgp 20263 df-rng 20277 df-ur 20310 df-ring 20363 df-subrg 20721 df-lmod 21035 df-lss 21105 df-sra 21346 df-rgmod 21347 df-dsmm 21934 df-frlm 21949

This theorem is used by: rrxplusgvscavalb 25607

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