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Mirrors > Home > MPE Home > Th. List > fzosplitprm1 | Structured version Visualization version GIF version |
Description: Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 25-Jun-2022.) |
Ref | Expression |
---|---|
fzosplitprm1 | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴..^(𝐵 + 1)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), 𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1135 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℤ) | |
2 | peano2zm 12658 | . . . . 5 ⊢ (𝐵 ∈ ℤ → (𝐵 − 1) ∈ ℤ) | |
3 | 2 | 3ad2ant2 1133 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐵 − 1) ∈ ℤ) |
4 | zltlem1 12668 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ 𝐴 ≤ (𝐵 − 1))) | |
5 | 4 | biimp3a 1468 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → 𝐴 ≤ (𝐵 − 1)) |
6 | eluz2 12882 | . . . 4 ⊢ ((𝐵 − 1) ∈ (ℤ≥‘𝐴) ↔ (𝐴 ∈ ℤ ∧ (𝐵 − 1) ∈ ℤ ∧ 𝐴 ≤ (𝐵 − 1))) | |
7 | 1, 3, 5, 6 | syl3anbrc 1342 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐵 − 1) ∈ (ℤ≥‘𝐴)) |
8 | fzosplitpr 13812 | . . 3 ⊢ ((𝐵 − 1) ∈ (ℤ≥‘𝐴) → (𝐴..^((𝐵 − 1) + 2)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), ((𝐵 − 1) + 1)})) | |
9 | 7, 8 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴..^((𝐵 − 1) + 2)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), ((𝐵 − 1) + 1)})) |
10 | zcn 12616 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ) | |
11 | 1cnd 11254 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → 1 ∈ ℂ) | |
12 | 2cnd 12342 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → 2 ∈ ℂ) | |
13 | 10, 11, 12 | subadd23d 11640 | . . . . . 6 ⊢ (𝐵 ∈ ℤ → ((𝐵 − 1) + 2) = (𝐵 + (2 − 1))) |
14 | 2m1e1 12390 | . . . . . . 7 ⊢ (2 − 1) = 1 | |
15 | 14 | oveq2i 7442 | . . . . . 6 ⊢ (𝐵 + (2 − 1)) = (𝐵 + 1) |
16 | 13, 15 | eqtr2di 2792 | . . . . 5 ⊢ (𝐵 ∈ ℤ → (𝐵 + 1) = ((𝐵 − 1) + 2)) |
17 | 16 | oveq2d 7447 | . . . 4 ⊢ (𝐵 ∈ ℤ → (𝐴..^(𝐵 + 1)) = (𝐴..^((𝐵 − 1) + 2))) |
18 | npcan1 11686 | . . . . . . . 8 ⊢ (𝐵 ∈ ℂ → ((𝐵 − 1) + 1) = 𝐵) | |
19 | 10, 18 | syl 17 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → ((𝐵 − 1) + 1) = 𝐵) |
20 | 19 | eqcomd 2741 | . . . . . 6 ⊢ (𝐵 ∈ ℤ → 𝐵 = ((𝐵 − 1) + 1)) |
21 | 20 | preq2d 4745 | . . . . 5 ⊢ (𝐵 ∈ ℤ → {(𝐵 − 1), 𝐵} = {(𝐵 − 1), ((𝐵 − 1) + 1)}) |
22 | 21 | uneq2d 4178 | . . . 4 ⊢ (𝐵 ∈ ℤ → ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), 𝐵}) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), ((𝐵 − 1) + 1)})) |
23 | 17, 22 | eqeq12d 2751 | . . 3 ⊢ (𝐵 ∈ ℤ → ((𝐴..^(𝐵 + 1)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), 𝐵}) ↔ (𝐴..^((𝐵 − 1) + 2)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), ((𝐵 − 1) + 1)}))) |
24 | 23 | 3ad2ant2 1133 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → ((𝐴..^(𝐵 + 1)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), 𝐵}) ↔ (𝐴..^((𝐵 − 1) + 2)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), ((𝐵 − 1) + 1)}))) |
25 | 9, 24 | mpbird 257 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴..^(𝐵 + 1)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), 𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 {cpr 4633 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 1c1 11154 + caddc 11156 < clt 11293 ≤ cle 11294 − cmin 11490 2c2 12319 ℤcz 12611 ℤ≥cuz 12876 ..^cfzo 13691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 |
This theorem is referenced by: (None) |
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