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 2903 | . 2 ⊢ (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ ran 𝐹) |
3 | ovex 7182 | . . . 4 ⊢ (𝑥 · 𝐴) ∈ V | |
4 | cycsubm.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴)) | |
5 | 3, 4 | fnmpti 6484 | . . 3 ⊢ 𝐹 Fn ℕ0 |
6 | fvelrnb 6719 | . . 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 7156 | . . . . . 6 ⊢ (𝑥 = 𝑖 → (𝑥 · 𝐴) = (𝑖 · 𝐴)) | |
9 | ovex 7182 | . . . . . 6 ⊢ (𝑖 · 𝐴) ∈ V | |
10 | 8, 4, 9 | fvmpt 6761 | . . . . 5 ⊢ (𝑖 ∈ ℕ0 → (𝐹‘𝑖) = (𝑖 · 𝐴)) |
11 | 10 | eqeq1d 2822 | . . . 4 ⊢ (𝑖 ∈ ℕ0 → ((𝐹‘𝑖) = 𝑋 ↔ (𝑖 · 𝐴) = 𝑋)) |
12 | eqcom 2827 | . . . 4 ⊢ ((𝑖 · 𝐴) = 𝑋 ↔ 𝑋 = (𝑖 · 𝐴)) | |
13 | 11, 12 | syl6bb 289 | . . 3 ⊢ (𝑖 ∈ ℕ0 → ((𝐹‘𝑖) = 𝑋 ↔ 𝑋 = (𝑖 · 𝐴))) |
14 | 13 | rexbiia 3245 | . 2 ⊢ (∃𝑖 ∈ ℕ0 (𝐹‘𝑖) = 𝑋 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴)) |
15 | 2, 7, 14 | 3bitri 299 | 1 ⊢ (𝑋 ∈ 𝐶 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1536 ∈ wcel 2113 ∃wrex 3138 ↦ cmpt 5139 ran crn 5549 Fn wfn 6343 ‘cfv 6348 (class class class)co 7149 ℕ0cn0 11891 Basecbs 16478 .gcmg 18219 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 df-ov 7152 |
This theorem is referenced by: cycsubmcl 18339 cycsubm 18340 cycsubmcom 18342 |
Copyright terms: Public domain | W3C validator |