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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 2881 | . 2 ⊢ (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ ran 𝐹) |
3 | ovex 7168 | . . . 4 ⊢ (𝑥 · 𝐴) ∈ V | |
4 | cycsubm.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴)) | |
5 | 3, 4 | fnmpti 6463 | . . 3 ⊢ 𝐹 Fn ℕ0 |
6 | fvelrnb 6701 | . . 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 7142 | . . . . . 6 ⊢ (𝑥 = 𝑖 → (𝑥 · 𝐴) = (𝑖 · 𝐴)) | |
9 | ovex 7168 | . . . . . 6 ⊢ (𝑖 · 𝐴) ∈ V | |
10 | 8, 4, 9 | fvmpt 6745 | . . . . 5 ⊢ (𝑖 ∈ ℕ0 → (𝐹‘𝑖) = (𝑖 · 𝐴)) |
11 | 10 | eqeq1d 2800 | . . . 4 ⊢ (𝑖 ∈ ℕ0 → ((𝐹‘𝑖) = 𝑋 ↔ (𝑖 · 𝐴) = 𝑋)) |
12 | eqcom 2805 | . . . 4 ⊢ ((𝑖 · 𝐴) = 𝑋 ↔ 𝑋 = (𝑖 · 𝐴)) | |
13 | 11, 12 | syl6bb 290 | . . 3 ⊢ (𝑖 ∈ ℕ0 → ((𝐹‘𝑖) = 𝑋 ↔ 𝑋 = (𝑖 · 𝐴))) |
14 | 13 | rexbiia 3209 | . 2 ⊢ (∃𝑖 ∈ ℕ0 (𝐹‘𝑖) = 𝑋 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴)) |
15 | 2, 7, 14 | 3bitri 300 | 1 ⊢ (𝑋 ∈ 𝐶 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ↦ cmpt 5110 ran crn 5520 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 ℕ0cn0 11885 Basecbs 16475 .gcmg 18216 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 df-ov 7138 |
This theorem is referenced by: cycsubmcl 18336 cycsubm 18337 cycsubmcom 18339 |
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