frlmvscavalb - Metamath Proof Explorer

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Theorem frlmvscavalb 20540

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 20529	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ (𝐾 ↑_m 𝐼))
7	1, 2, 6	syl2anc 587	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝐾 ↑_m 𝐼))
8	4	fvexi 6676	. . . . . . 7 ⊢ 𝐾 ∈ V
9	8	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ V)
10	9, 1	elmapd 8435	. . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝐾 ↑_m 𝐼) ↔ 𝑍:𝐼⟶𝐾))
11	7, 10	mpbid 235	. . . 4 ⊢ (𝜑 → 𝑍:𝐼⟶𝐾)
12	11	ffnd 6503	. . 3 ⊢ (𝜑 → 𝑍 Fn 𝐼)
13		frlmplusgvalb.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
14	3	frlmlmod 20519	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod)
15	13, 1, 14	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ LMod)
16		frlmvscavalb.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐾)
17	16, 4	eleqtrdi 2862	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅))
18	3	frlmsca 20523	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘𝐹))
19	13, 1, 18	syl2anc 587	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹))
20	19	fveq2d 6666	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
21	17, 20	eleqtrd 2854	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝐹)))
22		frlmplusgvalb.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
23		eqid 2758	. . . . . . . 8 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹)
24		frlmvscavalb.v	. . . . . . . 8 ⊢ ∙ = ( ·_𝑠 ‘𝐹)
25		eqid 2758	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
26	5, 23, 24, 25	lmodvscl 19724	. . . . . . 7 ⊢ ((𝐹 ∈ LMod ∧ 𝐴 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑋 ∈ 𝐵) → (𝐴 ∙ 𝑋) ∈ 𝐵)
27	15, 21, 22, 26	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (𝐴 ∙ 𝑋) ∈ 𝐵)
28	3, 4, 5	frlmbasmap 20529	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ (𝐴 ∙ 𝑋) ∈ 𝐵) → (𝐴 ∙ 𝑋) ∈ (𝐾 ↑_m 𝐼))
29	1, 27, 28	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝐴 ∙ 𝑋) ∈ (𝐾 ↑_m 𝐼))
30	9, 1	elmapd 8435	. . . . 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 6503	. . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) Fn 𝐼)
33		eqfnfv 6797	. . 3 ⊢ ((𝑍 Fn 𝐼 ∧ (𝐴 ∙ 𝑋) Fn 𝐼) → (𝑍 = (𝐴 ∙ 𝑋) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 ∙ 𝑋)‘𝑖)))
34	12, 32, 33	syl2anc 587	. 2 ⊢ (𝜑 → (𝑍 = (𝐴 ∙ 𝑋) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 ∙ 𝑋)‘𝑖)))
35	1	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ 𝑊)
36	16	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐴 ∈ 𝐾)
37	22	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑋 ∈ 𝐵)
38		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
39		frlmvscavalb.t	. . . . 5 ⊢ · = (._r‘𝑅)
40	3, 5, 4, 35, 36, 37, 38, 24, 39	frlmvscaval 20538	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐴 ∙ 𝑋)‘𝑖) = (𝐴 · (𝑋‘𝑖)))
41	40	eqeq2d 2769	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑍‘𝑖) = ((𝐴 ∙ 𝑋)‘𝑖) ↔ (𝑍‘𝑖) = (𝐴 · (𝑋‘𝑖))))
42	41	ralbidva 3125	. 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 399 = wceq 1538 ∈ wcel 2111 ∀wral 3070 Vcvv 3409 Fn wfn 6334 ⟶wf 6335 ‘cfv 6339 (class class class)co 7155 ↑_m cmap 8421 Basecbs 16546 ._rcmulr 16629 Scalarcsca 16631 ·_𝑠 cvsca 16632 Ringcrg 19370 LModclmod 19707 freeLMod cfrlm 20516

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5159 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 ax-cnex 10636 ax-resscn 10637 ax-1cn 10638 ax-icn 10639 ax-addcl 10640 ax-addrcl 10641 ax-mulcl 10642 ax-mulrcl 10643 ax-mulcom 10644 ax-addass 10645 ax-mulass 10646 ax-distr 10647 ax-i2m1 10648 ax-1ne0 10649 ax-1rid 10650 ax-rnegex 10651 ax-rrecex 10652 ax-cnre 10653 ax-pre-lttri 10654 ax-pre-lttrn 10655 ax-pre-ltadd 10656 ax-pre-mulgt0 10657

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7410 df-om 7585 df-1st 7698 df-2nd 7699 df-supp 7841 df-wrecs 7962 df-recs 8023 df-rdg 8061 df-1o 8117 df-er 8304 df-map 8423 df-ixp 8485 df-en 8533 df-dom 8534 df-sdom 8535 df-fin 8536 df-fsupp 8872 df-sup 8944 df-pnf 10720 df-mnf 10721 df-xr 10722 df-ltxr 10723 df-le 10724 df-sub 10915 df-neg 10916 df-nn 11680 df-2 11742 df-3 11743 df-4 11744 df-5 11745 df-6 11746 df-7 11747 df-8 11748 df-9 11749 df-n0 11940 df-z 12026 df-dec 12143 df-uz 12288 df-fz 12945 df-struct 16548 df-ndx 16549 df-slot 16550 df-base 16552 df-sets 16553 df-ress 16554 df-plusg 16641 df-mulr 16642 df-sca 16644 df-vsca 16645 df-ip 16646 df-tset 16647 df-ple 16648 df-ds 16650 df-hom 16652 df-cco 16653 df-0g 16778 df-prds 16784 df-pws 16786 df-mgm 17923 df-sgrp 17972 df-mnd 17983 df-grp 18177 df-minusg 18178 df-sbg 18179 df-subg 18348 df-mgp 19313 df-ur 19325 df-ring 19372 df-subrg 19606 df-lmod 19709 df-lss 19777 df-sra 20017 df-rgmod 20018 df-dsmm 20502 df-frlm 20517

This theorem is referenced by: (None)

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