frlmvplusgscavalb - Metamath Proof Explorer

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Theorem frlmvplusgscavalb 21890

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 21868	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod)
6	4, 3, 5	syl2anc 595	. . . 4 ⊢ (𝜑 → 𝐹 ∈ LMod)
7		frlmvscavalb.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐾)
8		frlmvscavalb.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑅)
9	7, 8	eleqtrdi 2879	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅))
10	1	frlmsca 21872	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘𝐹))
11	4, 3, 10	syl2anc 595	. . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹))
12	11	fveq2d 6886	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
13	9, 12	eleqtrd 2871	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝐹)))
14		frlmplusgvalb.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
15		eqid 2769	. . . . 5 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹)
16		frlmvscavalb.v	. . . . 5 ⊢ ∙ = ( ·_𝑠 ‘𝐹)
17		eqid 2769	. . . . 5 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
18	2, 15, 16, 17	lmodvscl 20977	. . . 4 ⊢ ((𝐹 ∈ LMod ∧ 𝐴 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑋 ∈ 𝐵) → (𝐴 ∙ 𝑋) ∈ 𝐵)
19	6, 13, 14, 18	syl3anc 1396	. . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) ∈ 𝐵)
20		frlmplusgvalb.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
21		frlmvplusgscavalb.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
22	21, 8	eleqtrdi 2879	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝑅))
23	22, 12	eleqtrd 2871	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (Base‘(Scalar‘𝐹)))
24		frlmvplusgscavalb.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
25	2, 15, 16, 17	lmodvscl 20977	. . . 4 ⊢ ((𝐹 ∈ LMod ∧ 𝐶 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑌 ∈ 𝐵) → (𝐶 ∙ 𝑌) ∈ 𝐵)
26	6, 23, 24, 25	syl3anc 1396	. . 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 21888	. 2 ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖))))
30	3	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ 𝑊)
31	7	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐴 ∈ 𝐾)
32	14	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑋 ∈ 𝐵)
33		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
34		frlmvscavalb.t	. . . . . 6 ⊢ · = (._r‘𝑅)
35	1, 2, 8, 30, 31, 32, 33, 16, 34	frlmvscaval 21887	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐴 ∙ 𝑋)‘𝑖) = (𝐴 · (𝑋‘𝑖)))
36	21	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐶 ∈ 𝐾)
37	24	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑌 ∈ 𝐵)
38	1, 2, 8, 30, 36, 37, 33, 16, 34	frlmvscaval 21887	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐶 ∙ 𝑌)‘𝑖) = (𝐶 · (𝑌‘𝑖)))
39	35, 38	oveq12d 7429	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖))))
40	39	eqeq2d 2780	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑍‘𝑖) = (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)) ↔ (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖)))))
41	40	ralbidva 3192	. 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

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ∀wral 3085 ‘cfv 6537 (class class class)co 7411 Basecbs 17269 +_gcplusg 17310 ._rcmulr 17311 Scalarcsca 17313 ·_𝑠 cvsca 17314 Ringcrg 20315 LModclmod 20959 freeLMod cfrlm 21865

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8694 df-map 8826 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-sup 9402 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-fz 13536 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-hom 17334 df-cco 17335 df-0g 17494 df-prds 17500 df-pws 17502 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-grp 19003 df-minusg 19004 df-sbg 19005 df-subg 19189 df-cmn 19852 df-abl 19853 df-mgp 20217 df-rng 20231 df-ur 20264 df-ring 20317 df-subrg 20655 df-lmod 20961 df-lss 21031 df-sra 21272 df-rgmod 21273 df-dsmm 21851 df-frlm 21866

This theorem is referenced by: rrxplusgvscavalb 25523

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