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Mirrors > Home > ILE Home > Th. List > fzrev3 | GIF version |
Description: The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.) |
Ref | Expression |
---|---|
fzrev3 | ⊢ (𝐾 ∈ ℤ → (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝐾 ∈ ℤ) | |
2 | elfzel1 9980 | . . . 4 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) | |
3 | 2 | adantl 275 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ) |
4 | elfzel2 9979 | . . . 4 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ) | |
5 | 4 | adantl 275 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝐾 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℤ) |
6 | 1, 3, 5 | 3jca 1172 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
7 | simpl 108 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁)) → 𝐾 ∈ ℤ) | |
8 | elfzel1 9980 | . . . 4 ⊢ (((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) | |
9 | 8 | adantl 275 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ) |
10 | elfzel2 9979 | . . . 4 ⊢ (((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ) | |
11 | 10 | adantl 275 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁)) → 𝑁 ∈ ℤ) |
12 | 7, 9, 11 | 3jca 1172 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁)) → (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
13 | zcn 9217 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
14 | zcn 9217 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
15 | pncan 8125 | . . . . . . 7 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑀 + 𝑁) − 𝑁) = 𝑀) | |
16 | pncan2 8126 | . . . . . . 7 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑀 + 𝑁) − 𝑀) = 𝑁) | |
17 | 15, 16 | oveq12d 5871 | . . . . . 6 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (((𝑀 + 𝑁) − 𝑁)...((𝑀 + 𝑁) − 𝑀)) = (𝑀...𝑁)) |
18 | 13, 14, 17 | syl2an 287 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑀 + 𝑁) − 𝑁)...((𝑀 + 𝑁) − 𝑀)) = (𝑀...𝑁)) |
19 | 18 | eleq2d 2240 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (((𝑀 + 𝑁) − 𝑁)...((𝑀 + 𝑁) − 𝑀)) ↔ 𝐾 ∈ (𝑀...𝑁))) |
20 | 19 | 3adant1 1010 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (((𝑀 + 𝑁) − 𝑁)...((𝑀 + 𝑁) − 𝑀)) ↔ 𝐾 ∈ (𝑀...𝑁))) |
21 | 3simpc 991 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) | |
22 | zaddcl 9252 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ) | |
23 | 22 | 3adant1 1010 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ) |
24 | simp1 992 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝐾 ∈ ℤ) | |
25 | fzrev 10040 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑀 + 𝑁) ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 ∈ (((𝑀 + 𝑁) − 𝑁)...((𝑀 + 𝑁) − 𝑀)) ↔ ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁))) | |
26 | 21, 23, 24, 25 | syl12anc 1231 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (((𝑀 + 𝑁) − 𝑁)...((𝑀 + 𝑁) − 𝑀)) ↔ ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁))) |
27 | 20, 26 | bitr3d 189 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁))) |
28 | 6, 12, 27 | pm5.21nd 911 | 1 ⊢ (𝐾 ∈ ℤ → (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 (class class class)co 5853 ℂcc 7772 + caddc 7777 − cmin 8090 ℤcz 9212 ...cfz 9965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-fz 9966 |
This theorem is referenced by: fzrev3i 10044 |
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