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Mirrors > Home > MPE Home > Th. List > elfzom1elp1fzo1 | Structured version Visualization version GIF version |
Description: Membership of a nonnegative integer incremented by one in a half-open range of positive integers. (Contributed by AV, 20-Mar-2021.) |
Ref | Expression |
---|---|
elfzom1elp1fzo1 | ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(𝑁 − 1))) → (𝐼 + 1) ∈ (1..^𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzoelz 13656 | . . . . . 6 ⊢ (𝐼 ∈ (0..^(𝑁 − 1)) → 𝐼 ∈ ℤ) | |
2 | 1 | zcnd 12689 | . . . . 5 ⊢ (𝐼 ∈ (0..^(𝑁 − 1)) → 𝐼 ∈ ℂ) |
3 | pncan1 11660 | . . . . 5 ⊢ (𝐼 ∈ ℂ → ((𝐼 + 1) − 1) = 𝐼) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐼 ∈ (0..^(𝑁 − 1)) → ((𝐼 + 1) − 1) = 𝐼) |
5 | id 22 | . . . 4 ⊢ (𝐼 ∈ (0..^(𝑁 − 1)) → 𝐼 ∈ (0..^(𝑁 − 1))) | |
6 | 4, 5 | eqeltrd 2828 | . . 3 ⊢ (𝐼 ∈ (0..^(𝑁 − 1)) → ((𝐼 + 1) − 1) ∈ (0..^(𝑁 − 1))) |
7 | 6 | adantl 481 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(𝑁 − 1))) → ((𝐼 + 1) − 1) ∈ (0..^(𝑁 − 1))) |
8 | 1 | peano2zd 12691 | . . . . 5 ⊢ (𝐼 ∈ (0..^(𝑁 − 1)) → (𝐼 + 1) ∈ ℤ) |
9 | 8 | anim1i 614 | . . . 4 ⊢ ((𝐼 ∈ (0..^(𝑁 − 1)) ∧ 𝑁 ∈ ℤ) → ((𝐼 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
10 | 9 | ancoms 458 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(𝑁 − 1))) → ((𝐼 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
11 | elfzom1b 13755 | . . 3 ⊢ (((𝐼 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐼 + 1) ∈ (1..^𝑁) ↔ ((𝐼 + 1) − 1) ∈ (0..^(𝑁 − 1)))) | |
12 | 10, 11 | syl 17 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(𝑁 − 1))) → ((𝐼 + 1) ∈ (1..^𝑁) ↔ ((𝐼 + 1) − 1) ∈ (0..^(𝑁 − 1)))) |
13 | 7, 12 | mpbird 257 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(𝑁 − 1))) → (𝐼 + 1) ∈ (1..^𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 (class class class)co 7414 ℂcc 11128 0cc0 11130 1c1 11131 + caddc 11133 − cmin 11466 ℤcz 12580 ..^cfzo 13651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-n0 12495 df-z 12581 df-uz 12845 df-fz 13509 df-fzo 13652 |
This theorem is referenced by: cshimadifsn0 14805 |
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