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| Mirrors > Home > MPE Home > Th. List > smndex1mndlem | Structured version Visualization version GIF version | ||
| Description: Lemma for smndex1mnd 18923 and smndex1id 18924. (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 4153 | . . 3 ⊢ (𝑋 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ↔ (𝑋 ∈ {𝐼} ∨ 𝑋 ∈ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})) | |
| 2 | elsni 4643 | . . . . 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 18914 | . . . . . . 7 ⊢ (𝐼 ∘ 𝐼) = 𝐼 |
| 7 | coeq2 5869 | . . . . . . 7 ⊢ (𝑋 = 𝐼 → (𝐼 ∘ 𝑋) = (𝐼 ∘ 𝐼)) | |
| 8 | id 22 | . . . . . . 7 ⊢ (𝑋 = 𝐼 → 𝑋 = 𝐼) | |
| 9 | 6, 7, 8 | 3eqtr4a 2803 | . . . . . 6 ⊢ (𝑋 = 𝐼 → (𝐼 ∘ 𝑋) = 𝑋) |
| 10 | coeq1 5868 | . . . . . . 7 ⊢ (𝑋 = 𝐼 → (𝑋 ∘ 𝐼) = (𝐼 ∘ 𝐼)) | |
| 11 | 6, 10, 8 | 3eqtr4a 2803 | . . . . . 6 ⊢ (𝑋 = 𝐼 → (𝑋 ∘ 𝐼) = 𝑋) |
| 12 | 9, 11 | jca 511 | . . . . 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 4995 | . . . . 5 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} ↔ ∃𝑛 ∈ (0..^𝑁)𝑋 ∈ {(𝐺‘𝑛)}) | |
| 15 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘)) | |
| 16 | 15 | sneqd 4638 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → {(𝐺‘𝑛)} = {(𝐺‘𝑘)}) |
| 17 | 16 | eleq2d 2827 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑋 ∈ {(𝐺‘𝑛)} ↔ 𝑋 ∈ {(𝐺‘𝑘)})) |
| 18 | 17 | cbvrexvw 3238 | . . . . . 6 ⊢ (∃𝑛 ∈ (0..^𝑁)𝑋 ∈ {(𝐺‘𝑛)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑋 ∈ {(𝐺‘𝑘)}) |
| 19 | elsni 4643 | . . . . . . . . 9 ⊢ (𝑋 ∈ {(𝐺‘𝑘)} → 𝑋 = (𝐺‘𝑘)) | |
| 20 | smndex1ibas.g | . . . . . . . . . . . 12 ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) | |
| 21 | 3, 4, 5, 20 | smndex1igid 18917 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ (0..^𝑁) → (𝐼 ∘ (𝐺‘𝑘)) = (𝐺‘𝑘)) |
| 22 | 3, 4, 5 | smndex1ibas 18913 | . . . . . . . . . . . 12 ⊢ 𝐼 ∈ (Base‘𝑀) |
| 23 | 3, 4, 5, 20 | smndex1gid 18916 | . . . . . . . . . . . 12 ⊢ ((𝐼 ∈ (Base‘𝑀) ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘)) |
| 24 | 22, 23 | mpan 690 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ (0..^𝑁) → ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘)) |
| 25 | 21, 24 | jca 511 | . . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑁) → ((𝐼 ∘ (𝐺‘𝑘)) = (𝐺‘𝑘) ∧ ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘))) |
| 26 | coeq2 5869 | . . . . . . . . . . . 12 ⊢ (𝑋 = (𝐺‘𝑘) → (𝐼 ∘ 𝑋) = (𝐼 ∘ (𝐺‘𝑘))) | |
| 27 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑋 = (𝐺‘𝑘) → 𝑋 = (𝐺‘𝑘)) | |
| 28 | 26, 27 | eqeq12d 2753 | . . . . . . . . . . 11 ⊢ (𝑋 = (𝐺‘𝑘) → ((𝐼 ∘ 𝑋) = 𝑋 ↔ (𝐼 ∘ (𝐺‘𝑘)) = (𝐺‘𝑘))) |
| 29 | coeq1 5868 | . . . . . . . . . . . 12 ⊢ (𝑋 = (𝐺‘𝑘) → (𝑋 ∘ 𝐼) = ((𝐺‘𝑘) ∘ 𝐼)) | |
| 30 | 29, 27 | eqeq12d 2753 | . . . . . . . . . . 11 ⊢ (𝑋 = (𝐺‘𝑘) → ((𝑋 ∘ 𝐼) = 𝑋 ↔ ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘))) |
| 31 | 28, 30 | anbi12d 632 | . . . . . . . . . 10 ⊢ (𝑋 = (𝐺‘𝑘) → (((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋) ↔ ((𝐼 ∘ (𝐺‘𝑘)) = (𝐺‘𝑘) ∧ ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘)))) |
| 32 | 25, 31 | imbitrrid 246 | . . . . . . . . 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 407 | . . . . . . 7 ⊢ ((𝑘 ∈ (0..^𝑁) ∧ 𝑋 ∈ {(𝐺‘𝑘)}) → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) |
| 35 | 34 | rexlimiva 3147 | . . . . . 6 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑋 ∈ {(𝐺‘𝑘)} → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) |
| 36 | 18, 35 | sylbi 217 | . . . . 5 ⊢ (∃𝑛 ∈ (0..^𝑁)𝑋 ∈ {(𝐺‘𝑛)} → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) |
| 37 | 14, 36 | sylbi 217 | . . . 4 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) |
| 38 | 13, 37 | jaoi 858 | . . 3 ⊢ ((𝑋 ∈ {𝐼} ∨ 𝑋 ∈ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) |
| 39 | 1, 38 | sylbi 217 | . 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 2859 | 1 ⊢ (𝑋 ∈ 𝐵 → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ∪ cun 3949 {csn 4626 ∪ ciun 4991 ↦ cmpt 5225 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℕcn 12266 ℕ0cn0 12526 ..^cfzo 13694 mod cmo 13909 Basecbs 17247 ↾s cress 17274 EndoFMndcefmnd 18881 |
| 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 ax-pre-sup 11233 |
| 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-rmo 3380 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-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-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 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-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-tset 17316 df-efmnd 18882 |
| This theorem is referenced by: smndex1mnd 18923 smndex1id 18924 |
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