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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 2825 | . 2 ⊢ (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ ran 𝐹) |
3 | ovex 7215 | . . . 4 ⊢ (𝑥 · 𝐴) ∈ V | |
4 | cycsubm.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴)) | |
5 | 3, 4 | fnmpti 6490 | . . 3 ⊢ 𝐹 Fn ℕ0 |
6 | fvelrnb 6742 | . . 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 7189 | . . . . . 6 ⊢ (𝑥 = 𝑖 → (𝑥 · 𝐴) = (𝑖 · 𝐴)) | |
9 | ovex 7215 | . . . . . 6 ⊢ (𝑖 · 𝐴) ∈ V | |
10 | 8, 4, 9 | fvmpt 6787 | . . . . 5 ⊢ (𝑖 ∈ ℕ0 → (𝐹‘𝑖) = (𝑖 · 𝐴)) |
11 | 10 | eqeq1d 2741 | . . . 4 ⊢ (𝑖 ∈ ℕ0 → ((𝐹‘𝑖) = 𝑋 ↔ (𝑖 · 𝐴) = 𝑋)) |
12 | eqcom 2746 | . . . 4 ⊢ ((𝑖 · 𝐴) = 𝑋 ↔ 𝑋 = (𝑖 · 𝐴)) | |
13 | 11, 12 | bitrdi 290 | . . 3 ⊢ (𝑖 ∈ ℕ0 → ((𝐹‘𝑖) = 𝑋 ↔ 𝑋 = (𝑖 · 𝐴))) |
14 | 13 | rexbiia 3161 | . 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 1542 ∈ wcel 2114 ∃wrex 3055 ↦ cmpt 5120 ran crn 5536 Fn wfn 6344 ‘cfv 6349 (class class class)co 7182 ℕ0cn0 11988 Basecbs 16598 .gcmg 18354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-iota 6307 df-fun 6351 df-fn 6352 df-fv 6357 df-ov 7185 |
This theorem is referenced by: cycsubmcl 18474 cycsubm 18475 cycsubmcom 18477 |
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