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| Mirrors > Home > MPE Home > Th. List > smndex1id | Structured version Visualization version GIF version | ||
| Description: The modulo function 𝐼 is the identity of the monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾). (Contributed by AV, 16-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)}) | 
| smndex1mgm.s | ⊢ 𝑆 = (𝑀 ↾s 𝐵) | 
| Ref | Expression | 
|---|---|
| smndex1id | ⊢ 𝐼 = (0g‘𝑆) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | smndex1ibas.i | . . . . . 6 ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) | |
| 2 | nn0ex 12534 | . . . . . . 7 ⊢ ℕ0 ∈ V | |
| 3 | 2 | mptex 7244 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) ∈ V | 
| 4 | 1, 3 | eqeltri 2836 | . . . . 5 ⊢ 𝐼 ∈ V | 
| 5 | 4 | snid 4661 | . . . 4 ⊢ 𝐼 ∈ {𝐼} | 
| 6 | elun1 4181 | . . . 4 ⊢ (𝐼 ∈ {𝐼} → 𝐼 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})) | |
| 7 | 5, 6 | ax-mp 5 | . . 3 ⊢ 𝐼 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) | 
| 8 | smndex1mgm.b | . . 3 ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) | |
| 9 | 7, 8 | eleqtrri 2839 | . 2 ⊢ 𝐼 ∈ 𝐵 | 
| 10 | smndex1ibas.m | . . . . . 6 ⊢ 𝑀 = (EndoFMnd‘ℕ0) | |
| 11 | smndex1ibas.n | . . . . . 6 ⊢ 𝑁 ∈ ℕ | |
| 12 | smndex1ibas.g | . . . . . 6 ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) | |
| 13 | smndex1mgm.s | . . . . . 6 ⊢ 𝑆 = (𝑀 ↾s 𝐵) | |
| 14 | 10, 11, 1, 12, 8, 13 | smndex1bas 18920 | . . . . 5 ⊢ (Base‘𝑆) = 𝐵 | 
| 15 | 14 | eqcomi 2745 | . . . 4 ⊢ 𝐵 = (Base‘𝑆) | 
| 16 | 15 | a1i 11 | . . 3 ⊢ (𝐼 ∈ 𝐵 → 𝐵 = (Base‘𝑆)) | 
| 17 | snex 5435 | . . . . . 6 ⊢ {𝐼} ∈ V | |
| 18 | ovex 7465 | . . . . . . 7 ⊢ (0..^𝑁) ∈ V | |
| 19 | snex 5435 | . . . . . . 7 ⊢ {(𝐺‘𝑛)} ∈ V | |
| 20 | 18, 19 | iunex 7994 | . . . . . 6 ⊢ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} ∈ V | 
| 21 | 17, 20 | unex 7765 | . . . . 5 ⊢ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ∈ V | 
| 22 | 8, 21 | eqeltri 2836 | . . . 4 ⊢ 𝐵 ∈ V | 
| 23 | eqid 2736 | . . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
| 24 | 13, 23 | ressplusg 17335 | . . . 4 ⊢ (𝐵 ∈ V → (+g‘𝑀) = (+g‘𝑆)) | 
| 25 | 22, 24 | mp1i 13 | . . 3 ⊢ (𝐼 ∈ 𝐵 → (+g‘𝑀) = (+g‘𝑆)) | 
| 26 | id 22 | . . 3 ⊢ (𝐼 ∈ 𝐵 → 𝐼 ∈ 𝐵) | |
| 27 | 10, 11, 1 | smndex1ibas 18914 | . . . . . 6 ⊢ 𝐼 ∈ (Base‘𝑀) | 
| 28 | 27 | a1i 11 | . . . . 5 ⊢ (𝐼 ∈ 𝐵 → 𝐼 ∈ (Base‘𝑀)) | 
| 29 | 10, 11, 1, 12, 8 | smndex1basss 18919 | . . . . . 6 ⊢ 𝐵 ⊆ (Base‘𝑀) | 
| 30 | 29 | sseli 3978 | . . . . 5 ⊢ (𝑎 ∈ 𝐵 → 𝑎 ∈ (Base‘𝑀)) | 
| 31 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
| 32 | 10, 31, 23 | efmndov 18895 | . . . . 5 ⊢ ((𝐼 ∈ (Base‘𝑀) ∧ 𝑎 ∈ (Base‘𝑀)) → (𝐼(+g‘𝑀)𝑎) = (𝐼 ∘ 𝑎)) | 
| 33 | 28, 30, 32 | syl2an 596 | . . . 4 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝐼(+g‘𝑀)𝑎) = (𝐼 ∘ 𝑎)) | 
| 34 | 10, 11, 1, 12, 8, 13 | smndex1mndlem 18923 | . . . . . 6 ⊢ (𝑎 ∈ 𝐵 → ((𝐼 ∘ 𝑎) = 𝑎 ∧ (𝑎 ∘ 𝐼) = 𝑎)) | 
| 35 | 34 | simpld 494 | . . . . 5 ⊢ (𝑎 ∈ 𝐵 → (𝐼 ∘ 𝑎) = 𝑎) | 
| 36 | 35 | adantl 481 | . . . 4 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝐼 ∘ 𝑎) = 𝑎) | 
| 37 | 33, 36 | eqtrd 2776 | . . 3 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝐼(+g‘𝑀)𝑎) = 𝑎) | 
| 38 | 10, 31, 23 | efmndov 18895 | . . . . 5 ⊢ ((𝑎 ∈ (Base‘𝑀) ∧ 𝐼 ∈ (Base‘𝑀)) → (𝑎(+g‘𝑀)𝐼) = (𝑎 ∘ 𝐼)) | 
| 39 | 30, 28, 38 | syl2anr 597 | . . . 4 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝑎(+g‘𝑀)𝐼) = (𝑎 ∘ 𝐼)) | 
| 40 | 34 | simprd 495 | . . . . 5 ⊢ (𝑎 ∈ 𝐵 → (𝑎 ∘ 𝐼) = 𝑎) | 
| 41 | 40 | adantl 481 | . . . 4 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝑎 ∘ 𝐼) = 𝑎) | 
| 42 | 39, 41 | eqtrd 2776 | . . 3 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝑎(+g‘𝑀)𝐼) = 𝑎) | 
| 43 | 16, 25, 26, 37, 42 | grpidd 18685 | . 2 ⊢ (𝐼 ∈ 𝐵 → 𝐼 = (0g‘𝑆)) | 
| 44 | 9, 43 | ax-mp 5 | 1 ⊢ 𝐼 = (0g‘𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∪ cun 3948 {csn 4625 ∪ ciun 4990 ↦ cmpt 5224 ∘ ccom 5688 ‘cfv 6560 (class class class)co 7432 0cc0 11156 ℕcn 12267 ℕ0cn0 12528 ..^cfzo 13695 mod cmo 13910 Basecbs 17248 ↾s cress 17275 +gcplusg 17298 0gc0g 17485 EndoFMndcefmnd 18882 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-tset 17317 df-0g 17487 df-efmnd 18883 | 
| This theorem is referenced by: (None) | 
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