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Theorem frlmvscavalb 20916

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 20905	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ (𝐾 ↑_m 𝐼))
7	1, 2, 6	syl2anc 586	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝐾 ↑_m 𝐼))
8	4	fvexi 6686	. . . . . . 7 ⊢ 𝐾 ∈ V
9	8	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ V)
10	9, 1	elmapd 8422	. . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝐾 ↑_m 𝐼) ↔ 𝑍:𝐼⟶𝐾))
11	7, 10	mpbid 234	. . . 4 ⊢ (𝜑 → 𝑍:𝐼⟶𝐾)
12	11	ffnd 6517	. . 3 ⊢ (𝜑 → 𝑍 Fn 𝐼)
13		frlmplusgvalb.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
14	3	frlmlmod 20895	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod)
15	13, 1, 14	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ LMod)
16		frlmvscavalb.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐾)
17	16, 4	eleqtrdi 2925	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅))
18	3	frlmsca 20899	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘𝐹))
19	13, 1, 18	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹))
20	19	fveq2d 6676	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
21	17, 20	eleqtrd 2917	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝐹)))
22		frlmplusgvalb.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
23		eqid 2823	. . . . . . . 8 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹)
24		frlmvscavalb.v	. . . . . . . 8 ⊢ ∙ = ( ·_𝑠 ‘𝐹)
25		eqid 2823	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
26	5, 23, 24, 25	lmodvscl 19653	. . . . . . 7 ⊢ ((𝐹 ∈ LMod ∧ 𝐴 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑋 ∈ 𝐵) → (𝐴 ∙ 𝑋) ∈ 𝐵)
27	15, 21, 22, 26	syl3anc 1367	. . . . . 6 ⊢ (𝜑 → (𝐴 ∙ 𝑋) ∈ 𝐵)
28	3, 4, 5	frlmbasmap 20905	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ (𝐴 ∙ 𝑋) ∈ 𝐵) → (𝐴 ∙ 𝑋) ∈ (𝐾 ↑_m 𝐼))
29	1, 27, 28	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝐴 ∙ 𝑋) ∈ (𝐾 ↑_m 𝐼))
30	9, 1	elmapd 8422	. . . . 5 ⊢ (𝜑 → ((𝐴 ∙ 𝑋) ∈ (𝐾 ↑_m 𝐼) ↔ (𝐴 ∙ 𝑋):𝐼⟶𝐾))
31	29, 30	mpbid 234	. . . 4 ⊢ (𝜑 → (𝐴 ∙ 𝑋):𝐼⟶𝐾)
32	31	ffnd 6517	. . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) Fn 𝐼)
33		eqfnfv 6804	. . 3 ⊢ ((𝑍 Fn 𝐼 ∧ (𝐴 ∙ 𝑋) Fn 𝐼) → (𝑍 = (𝐴 ∙ 𝑋) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 ∙ 𝑋)‘𝑖)))
34	12, 32, 33	syl2anc 586	. 2 ⊢ (𝜑 → (𝑍 = (𝐴 ∙ 𝑋) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 ∙ 𝑋)‘𝑖)))
35	1	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ 𝑊)
36	16	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐴 ∈ 𝐾)
37	22	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑋 ∈ 𝐵)
38		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
39		frlmvscavalb.t	. . . . 5 ⊢ · = (._r‘𝑅)
40	3, 5, 4, 35, 36, 37, 38, 24, 39	frlmvscaval 20914	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐴 ∙ 𝑋)‘𝑖) = (𝐴 · (𝑋‘𝑖)))
41	40	eqeq2d 2834	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑍‘𝑖) = ((𝐴 ∙ 𝑋)‘𝑖) ↔ (𝑍‘𝑖) = (𝐴 · (𝑋‘𝑖))))
42	41	ralbidva 3198	. 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 ∙ 𝑋)‘𝑖) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (𝐴 · (𝑋‘𝑖))))
43	34, 42	bitrd 281	1 ⊢ (𝜑 → (𝑍 = (𝐴 ∙ 𝑋) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (𝐴 · (𝑋‘𝑖))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑_m cmap 8408 Basecbs 16485 ._rcmulr 16568 Scalarcsca 16570 ·_𝑠 cvsca 16571 Ringcrg 19299 LModclmod 19636 freeLMod cfrlm 20892

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-hom 16591 df-cco 16592 df-0g 16717 df-prds 16723 df-pws 16725 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-mgp 19242 df-ur 19254 df-ring 19301 df-subrg 19535 df-lmod 19638 df-lss 19706 df-sra 19946 df-rgmod 19947 df-dsmm 20878 df-frlm 20893

This theorem is referenced by: (None)

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