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| Mirrors > Home > MPE Home > Th. List > Mathboxes > frlmfzolen | Structured version Visualization version GIF version | ||
| Description: The dimension of a vector of a module with indices from 0 to 𝑁 − 1. (Contributed by SN, 1-Sep-2023.) |
| Ref | Expression |
|---|---|
| frlmfzowrd.w | ⊢ 𝑊 = (𝐾 freeLMod (0..^𝑁)) |
| frlmfzowrd.b | ⊢ 𝐵 = (Base‘𝑊) |
| frlmfzowrd.s | ⊢ 𝑆 = (Base‘𝐾) |
| Ref | Expression |
|---|---|
| frlmfzolen | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (♯‘𝑋) = 𝑁) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ovexd 7466 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (0..^𝑁) ∈ V) | |
| 2 | frlmfzowrd.w | . . . 4 ⊢ 𝑊 = (𝐾 freeLMod (0..^𝑁)) | |
| 3 | frlmfzowrd.s | . . . 4 ⊢ 𝑆 = (Base‘𝐾) | |
| 4 | frlmfzowrd.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
| 5 | 2, 3, 4 | frlmbasf 21780 | . . 3 ⊢ (((0..^𝑁) ∈ V ∧ 𝑋 ∈ 𝐵) → 𝑋:(0..^𝑁)⟶𝑆) |
| 6 | 1, 5 | sylan 580 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → 𝑋:(0..^𝑁)⟶𝑆) |
| 7 | fnfzo0hash 14489 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋:(0..^𝑁)⟶𝑆) → (♯‘𝑋) = 𝑁) | |
| 8 | 6, 7 | syldan 591 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (♯‘𝑋) = 𝑁) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℕ0cn0 12526 ..^cfzo 13694 ♯chash 14369 Basecbs 17247 freeLMod cfrlm 21766 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-prds 17492 df-pws 17494 df-sra 21172 df-rgmod 21173 df-dsmm 21752 df-frlm 21767 |
| This theorem is referenced by: frlmfzowrdb 42514 |
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