frlmvplusgscavalb - Metamath Proof Explorer

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Theorem frlmvplusgscavalb 21738

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 21716	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod)
6	4, 3, 5	syl2anc 585	. . . 4 ⊢ (𝜑 → 𝐹 ∈ LMod)
7		frlmvscavalb.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐾)
8		frlmvscavalb.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑅)
9	7, 8	eleqtrdi 2847	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑅))
10	1	frlmsca 21720	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘𝐹))
11	4, 3, 10	syl2anc 585	. . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹))
12	11	fveq2d 6846	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
13	9, 12	eleqtrd 2839	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝐹)))
14		frlmplusgvalb.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
15		eqid 2737	. . . . 5 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹)
16		frlmvscavalb.v	. . . . 5 ⊢ ∙ = ( ·_𝑠 ‘𝐹)
17		eqid 2737	. . . . 5 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
18	2, 15, 16, 17	lmodvscl 20841	. . . 4 ⊢ ((𝐹 ∈ LMod ∧ 𝐴 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑋 ∈ 𝐵) → (𝐴 ∙ 𝑋) ∈ 𝐵)
19	6, 13, 14, 18	syl3anc 1374	. . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) ∈ 𝐵)
20		frlmplusgvalb.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
21		frlmvplusgscavalb.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
22	21, 8	eleqtrdi 2847	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝑅))
23	22, 12	eleqtrd 2839	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (Base‘(Scalar‘𝐹)))
24		frlmvplusgscavalb.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
25	2, 15, 16, 17	lmodvscl 20841	. . . 4 ⊢ ((𝐹 ∈ LMod ∧ 𝐶 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑌 ∈ 𝐵) → (𝐶 ∙ 𝑌) ∈ 𝐵)
26	6, 23, 24, 25	syl3anc 1374	. . 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 21736	. 2 ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖))))
30	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ 𝑊)
31	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐴 ∈ 𝐾)
32	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑋 ∈ 𝐵)
33		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
34		frlmvscavalb.t	. . . . . 6 ⊢ · = (._r‘𝑅)
35	1, 2, 8, 30, 31, 32, 33, 16, 34	frlmvscaval 21735	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐴 ∙ 𝑋)‘𝑖) = (𝐴 · (𝑋‘𝑖)))
36	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐶 ∈ 𝐾)
37	24	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑌 ∈ 𝐵)
38	1, 2, 8, 30, 36, 37, 33, 16, 34	frlmvscaval 21735	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐶 ∙ 𝑌)‘𝑖) = (𝐶 · (𝑌‘𝑖)))
39	35, 38	oveq12d 7386	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖))))
40	39	eqeq2d 2748	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑍‘𝑖) = (((𝐴 ∙ 𝑋)‘𝑖) + ((𝐶 ∙ 𝑌)‘𝑖)) ↔ (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖)))))
41	40	ralbidva 3159	. 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 279	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 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ‘cfv 6500 (class class class)co 7368 Basecbs 17148 +_gcplusg 17189 ._rcmulr 17190 Scalarcsca 17192 ·_𝑠 cvsca 17193 Ringcrg 20180 LModclmod 20823 freeLMod cfrlm 21713

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-hom 17213 df-cco 17214 df-0g 17373 df-prds 17379 df-pws 17381 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19065 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-subrg 20515 df-lmod 20825 df-lss 20895 df-sra 21137 df-rgmod 21138 df-dsmm 21699 df-frlm 21714

This theorem is referenced by: rrxplusgvscavalb 25363

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