Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2830 | . 2 ⊢ (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ ran 𝐹) |
3 | ovex 7308 | . . . 4 ⊢ (𝑥 · 𝐴) ∈ V | |
4 | cycsubm.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴)) | |
5 | 3, 4 | fnmpti 6576 | . . 3 ⊢ 𝐹 Fn ℕ0 |
6 | fvelrnb 6830 | . . 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 7282 | . . . . . 6 ⊢ (𝑥 = 𝑖 → (𝑥 · 𝐴) = (𝑖 · 𝐴)) | |
9 | ovex 7308 | . . . . . 6 ⊢ (𝑖 · 𝐴) ∈ V | |
10 | 8, 4, 9 | fvmpt 6875 | . . . . 5 ⊢ (𝑖 ∈ ℕ0 → (𝐹‘𝑖) = (𝑖 · 𝐴)) |
11 | 10 | eqeq1d 2740 | . . . 4 ⊢ (𝑖 ∈ ℕ0 → ((𝐹‘𝑖) = 𝑋 ↔ (𝑖 · 𝐴) = 𝑋)) |
12 | eqcom 2745 | . . . 4 ⊢ ((𝑖 · 𝐴) = 𝑋 ↔ 𝑋 = (𝑖 · 𝐴)) | |
13 | 11, 12 | bitrdi 287 | . . 3 ⊢ (𝑖 ∈ ℕ0 → ((𝐹‘𝑖) = 𝑋 ↔ 𝑋 = (𝑖 · 𝐴))) |
14 | 13 | rexbiia 3180 | . 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 205 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 ↦ cmpt 5157 ran crn 5590 Fn wfn 6428 ‘cfv 6433 (class class class)co 7275 ℕ0cn0 12233 Basecbs 16912 .gcmg 18700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6391 df-fun 6435 df-fn 6436 df-fv 6441 df-ov 7278 |
This theorem is referenced by: cycsubmcl 18820 cycsubm 18821 cycsubmcom 18823 |
Copyright terms: Public domain | W3C validator |