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Mirrors > Home > MPE Home > Th. List > smndex1basss | Structured version Visualization version GIF version |
Description: The modulo function 𝐼 and the constant functions (𝐺‘𝐾) are endofunctions on ℕ0. (Contributed by AV, 12-Feb-2024.) |
Ref | Expression |
---|---|
smndex1ibas.m | ⊢ 𝑀 = (EndoFMnd‘ℕ0) |
smndex1ibas.n | ⊢ 𝑁 ∈ ℕ |
smndex1ibas.i | ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) |
smndex1ibas.g | ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) |
smndex1mgm.b | ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) |
Ref | Expression |
---|---|
smndex1basss | ⊢ 𝐵 ⊆ (Base‘𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smndex1mgm.b | . . . . . 6 ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) | |
2 | 1 | eleq2i 2830 | . . . . 5 ⊢ (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})) |
3 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘)) | |
4 | 3 | sneqd 4642 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → {(𝐺‘𝑛)} = {(𝐺‘𝑘)}) |
5 | 4 | cbviunv 5044 | . . . . . . 7 ⊢ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} = ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} |
6 | 5 | uneq2i 4174 | . . . . . 6 ⊢ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) = ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) |
7 | 6 | eleq2i 2830 | . . . . 5 ⊢ (𝑏 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ↔ 𝑏 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)})) |
8 | 2, 7 | bitri 275 | . . . 4 ⊢ (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)})) |
9 | elun 4162 | . . . 4 ⊢ (𝑏 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ (𝑏 ∈ {𝐼} ∨ 𝑏 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)})) | |
10 | velsn 4646 | . . . . 5 ⊢ (𝑏 ∈ {𝐼} ↔ 𝑏 = 𝐼) | |
11 | eliun 4999 | . . . . 5 ⊢ (𝑏 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑏 ∈ {(𝐺‘𝑘)}) | |
12 | 10, 11 | orbi12i 914 | . . . 4 ⊢ ((𝑏 ∈ {𝐼} ∨ 𝑏 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ (𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 ∈ {(𝐺‘𝑘)})) |
13 | 8, 9, 12 | 3bitri 297 | . . 3 ⊢ (𝑏 ∈ 𝐵 ↔ (𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 ∈ {(𝐺‘𝑘)})) |
14 | smndex1ibas.m | . . . . . 6 ⊢ 𝑀 = (EndoFMnd‘ℕ0) | |
15 | smndex1ibas.n | . . . . . 6 ⊢ 𝑁 ∈ ℕ | |
16 | smndex1ibas.i | . . . . . 6 ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) | |
17 | 14, 15, 16 | smndex1ibas 18925 | . . . . 5 ⊢ 𝐼 ∈ (Base‘𝑀) |
18 | eleq1 2826 | . . . . 5 ⊢ (𝑏 = 𝐼 → (𝑏 ∈ (Base‘𝑀) ↔ 𝐼 ∈ (Base‘𝑀))) | |
19 | 17, 18 | mpbiri 258 | . . . 4 ⊢ (𝑏 = 𝐼 → 𝑏 ∈ (Base‘𝑀)) |
20 | smndex1ibas.g | . . . . . . . 8 ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) | |
21 | 14, 15, 16, 20 | smndex1gbas 18927 | . . . . . . 7 ⊢ (𝑘 ∈ (0..^𝑁) → (𝐺‘𝑘) ∈ (Base‘𝑀)) |
22 | 21 | adantr 480 | . . . . . 6 ⊢ ((𝑘 ∈ (0..^𝑁) ∧ 𝑏 ∈ {(𝐺‘𝑘)}) → (𝐺‘𝑘) ∈ (Base‘𝑀)) |
23 | elsni 4647 | . . . . . . . 8 ⊢ (𝑏 ∈ {(𝐺‘𝑘)} → 𝑏 = (𝐺‘𝑘)) | |
24 | 23 | eleq1d 2823 | . . . . . . 7 ⊢ (𝑏 ∈ {(𝐺‘𝑘)} → (𝑏 ∈ (Base‘𝑀) ↔ (𝐺‘𝑘) ∈ (Base‘𝑀))) |
25 | 24 | adantl 481 | . . . . . 6 ⊢ ((𝑘 ∈ (0..^𝑁) ∧ 𝑏 ∈ {(𝐺‘𝑘)}) → (𝑏 ∈ (Base‘𝑀) ↔ (𝐺‘𝑘) ∈ (Base‘𝑀))) |
26 | 22, 25 | mpbird 257 | . . . . 5 ⊢ ((𝑘 ∈ (0..^𝑁) ∧ 𝑏 ∈ {(𝐺‘𝑘)}) → 𝑏 ∈ (Base‘𝑀)) |
27 | 26 | rexlimiva 3144 | . . . 4 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑏 ∈ {(𝐺‘𝑘)} → 𝑏 ∈ (Base‘𝑀)) |
28 | 19, 27 | jaoi 857 | . . 3 ⊢ ((𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 ∈ {(𝐺‘𝑘)}) → 𝑏 ∈ (Base‘𝑀)) |
29 | 13, 28 | sylbi 217 | . 2 ⊢ (𝑏 ∈ 𝐵 → 𝑏 ∈ (Base‘𝑀)) |
30 | 29 | ssriv 3998 | 1 ⊢ 𝐵 ⊆ (Base‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 ∪ cun 3960 ⊆ wss 3962 {csn 4630 ∪ ciun 4995 ↦ cmpt 5230 ‘cfv 6562 (class class class)co 7430 0cc0 11152 ℕcn 12263 ℕ0cn0 12523 ..^cfzo 13690 mod cmo 13905 Basecbs 17244 EndoFMndcefmnd 18893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-tset 17316 df-efmnd 18894 |
This theorem is referenced by: smndex1bas 18931 smndex1mgm 18932 smndex1sgrp 18933 smndex1mnd 18935 smndex1id 18936 |
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