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Mirrors > Home > MPE Home > Th. List > frlmplusgvalb | Structured version Visualization version GIF version |
Description: Addition in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.) |
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) |
Ref | Expression |
---|---|
frlmplusgvalb | ⊢ (𝜑 → (𝑍 = (𝑋 ✚ 𝑌) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋‘𝑖) + (𝑌‘𝑖)))) |
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 2732 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
5 | frlmplusgvalb.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐹) | |
6 | 3, 4, 5 | frlmbasmap 21305 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ ((Base‘𝑅) ↑m 𝐼)) |
7 | 1, 2, 6 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ ((Base‘𝑅) ↑m 𝐼)) |
8 | fvexd 6903 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑅) ∈ V) | |
9 | 8, 1 | elmapd 8830 | . . . . 5 ⊢ (𝜑 → (𝑍 ∈ ((Base‘𝑅) ↑m 𝐼) ↔ 𝑍:𝐼⟶(Base‘𝑅))) |
10 | 7, 9 | mpbid 231 | . . . 4 ⊢ (𝜑 → 𝑍:𝐼⟶(Base‘𝑅)) |
11 | 10 | ffnd 6715 | . . 3 ⊢ (𝜑 → 𝑍 Fn 𝐼) |
12 | frlmplusgvalb.r | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
13 | 3 | frlmlmod 21295 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod) |
14 | 12, 1, 13 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ LMod) |
15 | lmodgrp 20470 | . . . . . . . 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 18823 | . . . . . . 7 ⊢ ((𝐹 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ✚ 𝑌) ∈ 𝐵) |
21 | 16, 17, 18, 20 | syl3anc 1371 | . . . . . 6 ⊢ (𝜑 → (𝑋 ✚ 𝑌) ∈ 𝐵) |
22 | 3, 4, 5 | frlmbasmap 21305 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ (𝑋 ✚ 𝑌) ∈ 𝐵) → (𝑋 ✚ 𝑌) ∈ ((Base‘𝑅) ↑m 𝐼)) |
23 | 1, 21, 22 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑋 ✚ 𝑌) ∈ ((Base‘𝑅) ↑m 𝐼)) |
24 | 8, 1 | elmapd 8830 | . . . . 5 ⊢ (𝜑 → ((𝑋 ✚ 𝑌) ∈ ((Base‘𝑅) ↑m 𝐼) ↔ (𝑋 ✚ 𝑌):𝐼⟶(Base‘𝑅))) |
25 | 23, 24 | mpbid 231 | . . . 4 ⊢ (𝜑 → (𝑋 ✚ 𝑌):𝐼⟶(Base‘𝑅)) |
26 | 25 | ffnd 6715 | . . 3 ⊢ (𝜑 → (𝑋 ✚ 𝑌) Fn 𝐼) |
27 | eqfnfv 7029 | . . 3 ⊢ ((𝑍 Fn 𝐼 ∧ (𝑋 ✚ 𝑌) Fn 𝐼) → (𝑍 = (𝑋 ✚ 𝑌) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋 ✚ 𝑌)‘𝑖))) | |
28 | 11, 26, 27 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑍 = (𝑋 ✚ 𝑌) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋 ✚ 𝑌)‘𝑖))) |
29 | 12 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑅 ∈ Ring) |
30 | 1 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ 𝑊) |
31 | 17 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑋 ∈ 𝐵) |
32 | 18 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑌 ∈ 𝐵) |
33 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼) | |
34 | frlmplusgvalb.a | . . . . 5 ⊢ + = (+g‘𝑅) | |
35 | 3, 5, 29, 30, 31, 32, 33, 34, 19 | frlmvplusgvalc 21313 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑋 ✚ 𝑌)‘𝑖) = ((𝑋‘𝑖) + (𝑌‘𝑖))) |
36 | 35 | eqeq2d 2743 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑍‘𝑖) = ((𝑋 ✚ 𝑌)‘𝑖) ↔ (𝑍‘𝑖) = ((𝑋‘𝑖) + (𝑌‘𝑖)))) |
37 | 36 | ralbidva 3175 | . 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋 ✚ 𝑌)‘𝑖) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋‘𝑖) + (𝑌‘𝑖)))) |
38 | 28, 37 | bitrd 278 | 1 ⊢ (𝜑 → (𝑍 = (𝑋 ✚ 𝑌) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋‘𝑖) + (𝑌‘𝑖)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑m cmap 8816 Basecbs 17140 +gcplusg 17193 Grpcgrp 18815 Ringcrg 20049 LModclmod 20463 freeLMod cfrlm 21292 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-hom 17217 df-cco 17218 df-0g 17383 df-prds 17389 df-pws 17391 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-mgp 19982 df-ur 19999 df-ring 20051 df-subrg 20353 df-lmod 20465 df-lss 20535 df-sra 20777 df-rgmod 20778 df-dsmm 21278 df-frlm 21293 |
This theorem is referenced by: frlmvplusgscavalb 21317 |
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