frlmvscavalb - Metamath Proof Explorer

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Theorem frlmvscavalb 21920

Description: 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)

**Assertion**
Ref	Expression
frlmvscavalb	⊢ (𝜑 → (𝑍 = (𝐴 ∙ 𝑋) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (𝐴 · (𝑋‘𝑖))))

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

**Proof of Theorem frlmvscavalb**
Step	Hyp	Ref	Expression
1		frlmplusgvalb.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
2		frlmplusgvalb.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
3		frlmplusgvalb.f	. . . . . . 7 ⊢ 𝐹 = (𝑅 freeLMod 𝐼)
4		frlmvscavalb.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝑅)
5		frlmplusgvalb.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐹)
6	3, 4, 5	frlmbasmap 21909	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ (𝐾 ↑_m 𝐼))
7	1, 2, 6	syl2anc 595	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝐾 ↑_m 𝐼))
8	4	fvexi 6895	. . . . . . 7 ⊢ 𝐾 ∈ V
9	8	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ V)
10	9, 1	elmapd 8833	. . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝐾 ↑_m 𝐼) ↔ 𝑍:𝐼⟶𝐾))
11	7, 10	mpbid 235	. . . 4 ⊢ (𝜑 → 𝑍:𝐼⟶𝐾)
12	11	ffnd 6706	. . 3 ⊢ (𝜑 → 𝑍 Fn 𝐼)
13		frlmplusgvalb.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
14	3	frlmlmod 21899	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod)
15	13, 1, 14	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ LMod)
16		frlmvscavalb.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐾)
17	16, 4	eleqtrdi 2873	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅))
18	3	frlmsca 21903	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘𝐹))
19	13, 1, 18	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹))
20	19	fveq2d 6885	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
21	17, 20	eleqtrd 2865	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝐹)))
22		frlmplusgvalb.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
23		eqid 2763	. . . . . . . 8 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹)
24		frlmvscavalb.v	. . . . . . . 8 ⊢ ∙ = ( ·_𝑠 ‘𝐹)
25		eqid 2763	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
26	5, 23, 24, 25	lmodvscl 20999	. . . . . . 7 ⊢ ((𝐹 ∈ LMod ∧ 𝐴 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑋 ∈ 𝐵) → (𝐴 ∙ 𝑋) ∈ 𝐵)
27	15, 21, 22, 26	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → (𝐴 ∙ 𝑋) ∈ 𝐵)
28	3, 4, 5	frlmbasmap 21909	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ (𝐴 ∙ 𝑋) ∈ 𝐵) → (𝐴 ∙ 𝑋) ∈ (𝐾 ↑_m 𝐼))
29	1, 27, 28	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝐴 ∙ 𝑋) ∈ (𝐾 ↑_m 𝐼))
30	9, 1	elmapd 8833	. . . . 5 ⊢ (𝜑 → ((𝐴 ∙ 𝑋) ∈ (𝐾 ↑_m 𝐼) ↔ (𝐴 ∙ 𝑋):𝐼⟶𝐾))
31	29, 30	mpbid 235	. . . 4 ⊢ (𝜑 → (𝐴 ∙ 𝑋):𝐼⟶𝐾)
32	31	ffnd 6706	. . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) Fn 𝐼)
33		eqfnfv 7025	. . 3 ⊢ ((𝑍 Fn 𝐼 ∧ (𝐴 ∙ 𝑋) Fn 𝐼) → (𝑍 = (𝐴 ∙ 𝑋) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 ∙ 𝑋)‘𝑖)))
34	12, 32, 33	syl2anc 595	. 2 ⊢ (𝜑 → (𝑍 = (𝐴 ∙ 𝑋) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 ∙ 𝑋)‘𝑖)))
35	1	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ 𝑊)
36	16	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐴 ∈ 𝐾)
37	22	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑋 ∈ 𝐵)
38		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
39		frlmvscavalb.t	. . . . 5 ⊢ · = (._r‘𝑅)
40	3, 5, 4, 35, 36, 37, 38, 24, 39	frlmvscaval 21918	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐴 ∙ 𝑋)‘𝑖) = (𝐴 · (𝑋‘𝑖)))
41	40	eqeq2d 2774	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑍‘𝑖) = ((𝐴 ∙ 𝑋)‘𝑖) ↔ (𝑍‘𝑖) = (𝐴 · (𝑋‘𝑖))))
42	41	ralbidva 3186	. 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 ∙ 𝑋)‘𝑖) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (𝐴 · (𝑋‘𝑖))))
43	34, 42	bitrd 282	1 ⊢ (𝜑 → (𝑍 = (𝐴 ∙ 𝑋) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (𝐴 · (𝑋‘𝑖))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∀wral 3079 Vcvv 3455 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ↑_m cmap 8820 Basecbs 17264 ._rcmulr 17306 Scalarcsca 17308 ·_𝑠 cvsca 17309 Ringcrg 20310 LModclmod 20981 freeLMod cfrlm 21896

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-sup 9398 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-fz 13531 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-hom 17329 df-cco 17330 df-0g 17489 df-prds 17495 df-pws 17497 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-grp 18998 df-minusg 18999 df-sbg 19000 df-subg 19184 df-cmn 19847 df-abl 19848 df-mgp 20212 df-rng 20226 df-ur 20259 df-ring 20312 df-subrg 20669 df-lmod 20983 df-lss 21053 df-sra 21294 df-rgmod 21295 df-dsmm 21882 df-frlm 21897

This theorem is referenced by: (None)

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