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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 2727 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
5 | frlmplusgvalb.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐹) | |
6 | 3, 4, 5 | frlmbasmap 21680 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ ((Base‘𝑅) ↑m 𝐼)) |
7 | 1, 2, 6 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ ((Base‘𝑅) ↑m 𝐼)) |
8 | fvexd 6906 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑅) ∈ V) | |
9 | 8, 1 | elmapd 8850 | . . . . 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 6717 | . . 3 ⊢ (𝜑 → 𝑍 Fn 𝐼) |
12 | frlmplusgvalb.r | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
13 | 3 | frlmlmod 21670 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod) |
14 | 12, 1, 13 | syl2anc 583 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ LMod) |
15 | lmodgrp 20739 | . . . . . . . 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 18889 | . . . . . . 7 ⊢ ((𝐹 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ✚ 𝑌) ∈ 𝐵) |
21 | 16, 17, 18, 20 | syl3anc 1369 | . . . . . 6 ⊢ (𝜑 → (𝑋 ✚ 𝑌) ∈ 𝐵) |
22 | 3, 4, 5 | frlmbasmap 21680 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ (𝑋 ✚ 𝑌) ∈ 𝐵) → (𝑋 ✚ 𝑌) ∈ ((Base‘𝑅) ↑m 𝐼)) |
23 | 1, 21, 22 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (𝑋 ✚ 𝑌) ∈ ((Base‘𝑅) ↑m 𝐼)) |
24 | 8, 1 | elmapd 8850 | . . . . 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 6717 | . . 3 ⊢ (𝜑 → (𝑋 ✚ 𝑌) Fn 𝐼) |
27 | eqfnfv 7034 | . . 3 ⊢ ((𝑍 Fn 𝐼 ∧ (𝑋 ✚ 𝑌) Fn 𝐼) → (𝑍 = (𝑋 ✚ 𝑌) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋 ✚ 𝑌)‘𝑖))) | |
28 | 11, 26, 27 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝑍 = (𝑋 ✚ 𝑌) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋 ✚ 𝑌)‘𝑖))) |
29 | 12 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑅 ∈ Ring) |
30 | 1 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ 𝑊) |
31 | 17 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑋 ∈ 𝐵) |
32 | 18 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑌 ∈ 𝐵) |
33 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼) | |
34 | frlmplusgvalb.a | . . . . 5 ⊢ + = (+g‘𝑅) | |
35 | 3, 5, 29, 30, 31, 32, 33, 34, 19 | frlmvplusgvalc 21688 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑋 ✚ 𝑌)‘𝑖) = ((𝑋‘𝑖) + (𝑌‘𝑖))) |
36 | 35 | eqeq2d 2738 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑍‘𝑖) = ((𝑋 ✚ 𝑌)‘𝑖) ↔ (𝑍‘𝑖) = ((𝑋‘𝑖) + (𝑌‘𝑖)))) |
37 | 36 | ralbidva 3170 | . 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋 ✚ 𝑌)‘𝑖) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋‘𝑖) + (𝑌‘𝑖)))) |
38 | 28, 37 | bitrd 279 | 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 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 Vcvv 3469 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ↑m cmap 8836 Basecbs 17171 +gcplusg 17224 Grpcgrp 18881 Ringcrg 20164 LModclmod 20732 freeLMod cfrlm 21667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-hom 17248 df-cco 17249 df-0g 17414 df-prds 17420 df-pws 17422 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-grp 18884 df-minusg 18885 df-sbg 18886 df-subg 19069 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-subrg 20497 df-lmod 20734 df-lss 20805 df-sra 21047 df-rgmod 21048 df-dsmm 21653 df-frlm 21668 |
This theorem is referenced by: frlmvplusgscavalb 21692 |
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