![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > smndex1mndlem | Structured version Visualization version GIF version |
Description: Lemma for smndex1mnd 18720 and smndex1id 18721. (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 |
---|---|
smndex1mndlem | ⊢ (𝑋 ∈ 𝐵 → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun 4108 | . . 3 ⊢ (𝑋 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ↔ (𝑋 ∈ {𝐼} ∨ 𝑋 ∈ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})) | |
2 | elsni 4603 | . . . . 5 ⊢ (𝑋 ∈ {𝐼} → 𝑋 = 𝐼) | |
3 | smndex1ibas.m | . . . . . . . 8 ⊢ 𝑀 = (EndoFMnd‘ℕ0) | |
4 | smndex1ibas.n | . . . . . . . 8 ⊢ 𝑁 ∈ ℕ | |
5 | smndex1ibas.i | . . . . . . . 8 ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) | |
6 | 3, 4, 5 | smndex1iidm 18711 | . . . . . . 7 ⊢ (𝐼 ∘ 𝐼) = 𝐼 |
7 | coeq2 5814 | . . . . . . 7 ⊢ (𝑋 = 𝐼 → (𝐼 ∘ 𝑋) = (𝐼 ∘ 𝐼)) | |
8 | id 22 | . . . . . . 7 ⊢ (𝑋 = 𝐼 → 𝑋 = 𝐼) | |
9 | 6, 7, 8 | 3eqtr4a 2802 | . . . . . 6 ⊢ (𝑋 = 𝐼 → (𝐼 ∘ 𝑋) = 𝑋) |
10 | coeq1 5813 | . . . . . . 7 ⊢ (𝑋 = 𝐼 → (𝑋 ∘ 𝐼) = (𝐼 ∘ 𝐼)) | |
11 | 6, 10, 8 | 3eqtr4a 2802 | . . . . . 6 ⊢ (𝑋 = 𝐼 → (𝑋 ∘ 𝐼) = 𝑋) |
12 | 9, 11 | jca 512 | . . . . 5 ⊢ (𝑋 = 𝐼 → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) |
13 | 2, 12 | syl 17 | . . . 4 ⊢ (𝑋 ∈ {𝐼} → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) |
14 | eliun 4958 | . . . . 5 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} ↔ ∃𝑛 ∈ (0..^𝑁)𝑋 ∈ {(𝐺‘𝑛)}) | |
15 | fveq2 6842 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘)) | |
16 | 15 | sneqd 4598 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → {(𝐺‘𝑛)} = {(𝐺‘𝑘)}) |
17 | 16 | eleq2d 2823 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑋 ∈ {(𝐺‘𝑛)} ↔ 𝑋 ∈ {(𝐺‘𝑘)})) |
18 | 17 | cbvrexvw 3226 | . . . . . 6 ⊢ (∃𝑛 ∈ (0..^𝑁)𝑋 ∈ {(𝐺‘𝑛)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑋 ∈ {(𝐺‘𝑘)}) |
19 | elsni 4603 | . . . . . . . . 9 ⊢ (𝑋 ∈ {(𝐺‘𝑘)} → 𝑋 = (𝐺‘𝑘)) | |
20 | smndex1ibas.g | . . . . . . . . . . . 12 ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) | |
21 | 3, 4, 5, 20 | smndex1igid 18714 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ (0..^𝑁) → (𝐼 ∘ (𝐺‘𝑘)) = (𝐺‘𝑘)) |
22 | 3, 4, 5 | smndex1ibas 18710 | . . . . . . . . . . . 12 ⊢ 𝐼 ∈ (Base‘𝑀) |
23 | 3, 4, 5, 20 | smndex1gid 18713 | . . . . . . . . . . . 12 ⊢ ((𝐼 ∈ (Base‘𝑀) ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘)) |
24 | 22, 23 | mpan 688 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ (0..^𝑁) → ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘)) |
25 | 21, 24 | jca 512 | . . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑁) → ((𝐼 ∘ (𝐺‘𝑘)) = (𝐺‘𝑘) ∧ ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘))) |
26 | coeq2 5814 | . . . . . . . . . . . 12 ⊢ (𝑋 = (𝐺‘𝑘) → (𝐼 ∘ 𝑋) = (𝐼 ∘ (𝐺‘𝑘))) | |
27 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑋 = (𝐺‘𝑘) → 𝑋 = (𝐺‘𝑘)) | |
28 | 26, 27 | eqeq12d 2752 | . . . . . . . . . . 11 ⊢ (𝑋 = (𝐺‘𝑘) → ((𝐼 ∘ 𝑋) = 𝑋 ↔ (𝐼 ∘ (𝐺‘𝑘)) = (𝐺‘𝑘))) |
29 | coeq1 5813 | . . . . . . . . . . . 12 ⊢ (𝑋 = (𝐺‘𝑘) → (𝑋 ∘ 𝐼) = ((𝐺‘𝑘) ∘ 𝐼)) | |
30 | 29, 27 | eqeq12d 2752 | . . . . . . . . . . 11 ⊢ (𝑋 = (𝐺‘𝑘) → ((𝑋 ∘ 𝐼) = 𝑋 ↔ ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘))) |
31 | 28, 30 | anbi12d 631 | . . . . . . . . . 10 ⊢ (𝑋 = (𝐺‘𝑘) → (((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋) ↔ ((𝐼 ∘ (𝐺‘𝑘)) = (𝐺‘𝑘) ∧ ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘)))) |
32 | 25, 31 | syl5ibr 245 | . . . . . . . . 9 ⊢ (𝑋 = (𝐺‘𝑘) → (𝑘 ∈ (0..^𝑁) → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋))) |
33 | 19, 32 | syl 17 | . . . . . . . 8 ⊢ (𝑋 ∈ {(𝐺‘𝑘)} → (𝑘 ∈ (0..^𝑁) → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋))) |
34 | 33 | impcom 408 | . . . . . . 7 ⊢ ((𝑘 ∈ (0..^𝑁) ∧ 𝑋 ∈ {(𝐺‘𝑘)}) → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) |
35 | 34 | rexlimiva 3144 | . . . . . 6 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑋 ∈ {(𝐺‘𝑘)} → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) |
36 | 18, 35 | sylbi 216 | . . . . 5 ⊢ (∃𝑛 ∈ (0..^𝑁)𝑋 ∈ {(𝐺‘𝑛)} → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) |
37 | 14, 36 | sylbi 216 | . . . 4 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) |
38 | 13, 37 | jaoi 855 | . . 3 ⊢ ((𝑋 ∈ {𝐼} ∨ 𝑋 ∈ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) |
39 | 1, 38 | sylbi 216 | . 2 ⊢ (𝑋 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) |
40 | smndex1mgm.b | . 2 ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) | |
41 | 39, 40 | eleq2s 2856 | 1 ⊢ (𝑋 ∈ 𝐵 → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∃wrex 3073 ∪ cun 3908 {csn 4586 ∪ ciun 4954 ↦ cmpt 5188 ∘ ccom 5637 ‘cfv 6496 (class class class)co 7357 0cc0 11051 ℕcn 12153 ℕ0cn0 12413 ..^cfzo 13567 mod cmo 13774 Basecbs 17083 ↾s cress 17112 EndoFMndcefmnd 18678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-struct 17019 df-slot 17054 df-ndx 17066 df-base 17084 df-plusg 17146 df-tset 17152 df-efmnd 18679 |
This theorem is referenced by: smndex1mnd 18720 smndex1id 18721 |
Copyright terms: Public domain | W3C validator |