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Theorem frlmplusgvalb 20731

Description: Addition 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)
frlmplusgvalb.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
frlmplusgvalb.a	⊢ + = (+_g‘𝑅)
frlmplusgvalb.p	⊢ ✚ = (+_g‘𝐹)

**Assertion**
Ref	Expression
frlmplusgvalb	⊢ (𝜑 → (𝑍 = (𝑋 ✚ 𝑌) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋‘𝑖) + (𝑌‘𝑖))))

Distinct variable groups: 𝑖,𝐼 𝑖,𝑋 𝑖,𝑍 𝜑,𝑖 𝑖,𝑌 ✚ ,𝑖 Allowed substitution hints: 𝐵(𝑖) + (𝑖) 𝑅(𝑖) 𝐹(𝑖) 𝑊(𝑖)

**Proof of Theorem frlmplusgvalb**
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		eqid 2737	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		frlmplusgvalb.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐹)
6	3, 4, 5	frlmbasmap 20721	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ ((Base‘𝑅) ↑_m 𝐼))
7	1, 2, 6	syl2anc 587	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ ((Base‘𝑅) ↑_m 𝐼))
8		fvexd 6732	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
9	8, 1	elmapd 8522	. . . . 5 ⊢ (𝜑 → (𝑍 ∈ ((Base‘𝑅) ↑_m 𝐼) ↔ 𝑍:𝐼⟶(Base‘𝑅)))
10	7, 9	mpbid 235	. . . 4 ⊢ (𝜑 → 𝑍:𝐼⟶(Base‘𝑅))
11	10	ffnd 6546	. . 3 ⊢ (𝜑 → 𝑍 Fn 𝐼)
12		frlmplusgvalb.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring)
13	3	frlmlmod 20711	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod)
14	12, 1, 13	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ LMod)
15		lmodgrp 19906	. . . . . . . 8 ⊢ (𝐹 ∈ LMod → 𝐹 ∈ Grp)
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ Grp)
17		frlmplusgvalb.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
18		frlmplusgvalb.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
19		frlmplusgvalb.p	. . . . . . . 8 ⊢ ✚ = (+_g‘𝐹)
20	5, 19	grpcl 18373	. . . . . . 7 ⊢ ((𝐹 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ✚ 𝑌) ∈ 𝐵)
21	16, 17, 18, 20	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝑋 ✚ 𝑌) ∈ 𝐵)
22	3, 4, 5	frlmbasmap 20721	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ (𝑋 ✚ 𝑌) ∈ 𝐵) → (𝑋 ✚ 𝑌) ∈ ((Base‘𝑅) ↑_m 𝐼))
23	1, 21, 22	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝑋 ✚ 𝑌) ∈ ((Base‘𝑅) ↑_m 𝐼))
24	8, 1	elmapd 8522	. . . . 5 ⊢ (𝜑 → ((𝑋 ✚ 𝑌) ∈ ((Base‘𝑅) ↑_m 𝐼) ↔ (𝑋 ✚ 𝑌):𝐼⟶(Base‘𝑅)))
25	23, 24	mpbid 235	. . . 4 ⊢ (𝜑 → (𝑋 ✚ 𝑌):𝐼⟶(Base‘𝑅))
26	25	ffnd 6546	. . 3 ⊢ (𝜑 → (𝑋 ✚ 𝑌) Fn 𝐼)
27		eqfnfv 6852	. . 3 ⊢ ((𝑍 Fn 𝐼 ∧ (𝑋 ✚ 𝑌) Fn 𝐼) → (𝑍 = (𝑋 ✚ 𝑌) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋 ✚ 𝑌)‘𝑖)))
28	11, 26, 27	syl2anc 587	. 2 ⊢ (𝜑 → (𝑍 = (𝑋 ✚ 𝑌) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋 ✚ 𝑌)‘𝑖)))
29	12	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑅 ∈ Ring)
30	1	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ 𝑊)
31	17	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑋 ∈ 𝐵)
32	18	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑌 ∈ 𝐵)
33		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
34		frlmplusgvalb.a	. . . . 5 ⊢ + = (+_g‘𝑅)
35	3, 5, 29, 30, 31, 32, 33, 34, 19	frlmvplusgvalc 20729	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑋 ✚ 𝑌)‘𝑖) = ((𝑋‘𝑖) + (𝑌‘𝑖)))
36	35	eqeq2d 2748	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑍‘𝑖) = ((𝑋 ✚ 𝑌)‘𝑖) ↔ (𝑍‘𝑖) = ((𝑋‘𝑖) + (𝑌‘𝑖))))
37	36	ralbidva 3117	. 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋 ✚ 𝑌)‘𝑖) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋‘𝑖) + (𝑌‘𝑖))))
38	28, 37	bitrd 282	1 ⊢ (𝜑 → (𝑍 = (𝑋 ✚ 𝑌) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋‘𝑖) + (𝑌‘𝑖))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 Vcvv 3408 Fn wfn 6375 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ↑_m cmap 8508 Basecbs 16760 +_gcplusg 16802 Grpcgrp 18365 Ringcrg 19562 LModclmod 19899 freeLMod cfrlm 20708

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-sup 9058 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-fz 13096 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-hom 16826 df-cco 16827 df-0g 16946 df-prds 16952 df-pws 16954 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-grp 18368 df-minusg 18369 df-sbg 18370 df-subg 18540 df-mgp 19505 df-ur 19517 df-ring 19564 df-subrg 19798 df-lmod 19901 df-lss 19969 df-sra 20209 df-rgmod 20210 df-dsmm 20694 df-frlm 20709

This theorem is referenced by: frlmvplusgscavalb 20733

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