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| Mirrors > Home > MPE Home > Th. List > cycsubmel | Structured version Visualization version GIF version | ||
| Description: Characterization of an element of the set of nonnegative integer powers of an element 𝐴. Although this theorem holds for any class 𝐺, the definition of 𝐹 is only meaningful if 𝐺 is a monoid or at least a unital magma. (Contributed by AV, 28-Dec-2023.) | 
| Ref | Expression | 
|---|---|
| cycsubm.b | ⊢ 𝐵 = (Base‘𝐺) | 
| cycsubm.t | ⊢ · = (.g‘𝐺) | 
| cycsubm.f | ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴)) | 
| cycsubm.c | ⊢ 𝐶 = ran 𝐹 | 
| Ref | Expression | 
|---|---|
| cycsubmel | ⊢ (𝑋 ∈ 𝐶 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cycsubm.c | . . 3 ⊢ 𝐶 = ran 𝐹 | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ ran 𝐹) | 
| 3 | ovex 7464 | . . . 4 ⊢ (𝑥 · 𝐴) ∈ V | |
| 4 | cycsubm.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴)) | |
| 5 | 3, 4 | fnmpti 6711 | . . 3 ⊢ 𝐹 Fn ℕ0 | 
| 6 | fvelrnb 6969 | . . 3 ⊢ (𝐹 Fn ℕ0 → (𝑋 ∈ ran 𝐹 ↔ ∃𝑖 ∈ ℕ0 (𝐹‘𝑖) = 𝑋)) | |
| 7 | 5, 6 | ax-mp 5 | . 2 ⊢ (𝑋 ∈ ran 𝐹 ↔ ∃𝑖 ∈ ℕ0 (𝐹‘𝑖) = 𝑋) | 
| 8 | oveq1 7438 | . . . . . 6 ⊢ (𝑥 = 𝑖 → (𝑥 · 𝐴) = (𝑖 · 𝐴)) | |
| 9 | ovex 7464 | . . . . . 6 ⊢ (𝑖 · 𝐴) ∈ V | |
| 10 | 8, 4, 9 | fvmpt 7016 | . . . . 5 ⊢ (𝑖 ∈ ℕ0 → (𝐹‘𝑖) = (𝑖 · 𝐴)) | 
| 11 | 10 | eqeq1d 2739 | . . . 4 ⊢ (𝑖 ∈ ℕ0 → ((𝐹‘𝑖) = 𝑋 ↔ (𝑖 · 𝐴) = 𝑋)) | 
| 12 | eqcom 2744 | . . . 4 ⊢ ((𝑖 · 𝐴) = 𝑋 ↔ 𝑋 = (𝑖 · 𝐴)) | |
| 13 | 11, 12 | bitrdi 287 | . . 3 ⊢ (𝑖 ∈ ℕ0 → ((𝐹‘𝑖) = 𝑋 ↔ 𝑋 = (𝑖 · 𝐴))) | 
| 14 | 13 | rexbiia 3092 | . 2 ⊢ (∃𝑖 ∈ ℕ0 (𝐹‘𝑖) = 𝑋 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴)) | 
| 15 | 2, 7, 14 | 3bitri 297 | 1 ⊢ (𝑋 ∈ 𝐶 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ↦ cmpt 5225 ran crn 5686 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 ℕ0cn0 12526 Basecbs 17247 .gcmg 19085 | 
| 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-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-ov 7434 | 
| This theorem is referenced by: cycsubmcl 19219 cycsubm 19220 cycsubmcom 19222 | 
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