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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 2824 | . . . . 5 ⊢ (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})) |
3 | fveq2 6891 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘)) | |
4 | 3 | sneqd 4640 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → {(𝐺‘𝑛)} = {(𝐺‘𝑘)}) |
5 | 4 | cbviunv 5043 | . . . . . . 7 ⊢ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} = ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} |
6 | 5 | uneq2i 4160 | . . . . . 6 ⊢ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) = ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) |
7 | 6 | eleq2i 2824 | . . . . 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 4148 | . . . 4 ⊢ (𝑏 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ (𝑏 ∈ {𝐼} ∨ 𝑏 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)})) | |
10 | velsn 4644 | . . . . 5 ⊢ (𝑏 ∈ {𝐼} ↔ 𝑏 = 𝐼) | |
11 | eliun 5001 | . . . . 5 ⊢ (𝑏 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑏 ∈ {(𝐺‘𝑘)}) | |
12 | 10, 11 | orbi12i 912 | . . . 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 18820 | . . . . 5 ⊢ 𝐼 ∈ (Base‘𝑀) |
18 | eleq1 2820 | . . . . 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 18822 | . . . . . . 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 4645 | . . . . . . . 8 ⊢ (𝑏 ∈ {(𝐺‘𝑘)} → 𝑏 = (𝐺‘𝑘)) | |
24 | 23 | eleq1d 2817 | . . . . . . 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 3146 | . . . 4 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑏 ∈ {(𝐺‘𝑘)} → 𝑏 ∈ (Base‘𝑀)) |
28 | 19, 27 | jaoi 854 | . . 3 ⊢ ((𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 ∈ {(𝐺‘𝑘)}) → 𝑏 ∈ (Base‘𝑀)) |
29 | 13, 28 | sylbi 216 | . 2 ⊢ (𝑏 ∈ 𝐵 → 𝑏 ∈ (Base‘𝑀)) |
30 | 29 | ssriv 3986 | 1 ⊢ 𝐵 ⊆ (Base‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 ∪ cun 3946 ⊆ wss 3948 {csn 4628 ∪ ciun 4997 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7412 0cc0 11116 ℕcn 12219 ℕ0cn0 12479 ..^cfzo 13634 mod cmo 13841 Basecbs 17151 EndoFMndcefmnd 18788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-fz 13492 df-fzo 13635 df-fl 13764 df-mod 13842 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-tset 17223 df-efmnd 18789 |
This theorem is referenced by: smndex1bas 18826 smndex1mgm 18827 smndex1sgrp 18828 smndex1mnd 18830 smndex1id 18831 |
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