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Mirrors > Home > MPE Home > Th. List > fzosplitsn | Structured version Visualization version GIF version |
Description: Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
Ref | Expression |
---|---|
fzosplitsn | ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴..^(𝐵 + 1)) = ((𝐴..^𝐵) ∪ {𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ (ℤ≥‘𝐴)) | |
2 | eluzelz 12778 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℤ) | |
3 | uzid 12783 | . . . . 5 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ (ℤ≥‘𝐵)) | |
4 | peano2uz 12831 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐵) → (𝐵 + 1) ∈ (ℤ≥‘𝐵)) | |
5 | 2, 3, 4 | 3syl 18 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 + 1) ∈ (ℤ≥‘𝐵)) |
6 | elfzuzb 13441 | . . . 4 ⊢ (𝐵 ∈ (𝐴...(𝐵 + 1)) ↔ (𝐵 ∈ (ℤ≥‘𝐴) ∧ (𝐵 + 1) ∈ (ℤ≥‘𝐵))) | |
7 | 1, 5, 6 | sylanbrc 584 | . . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ (𝐴...(𝐵 + 1))) |
8 | fzosplit 13611 | . . 3 ⊢ (𝐵 ∈ (𝐴...(𝐵 + 1)) → (𝐴..^(𝐵 + 1)) = ((𝐴..^𝐵) ∪ (𝐵..^(𝐵 + 1)))) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴..^(𝐵 + 1)) = ((𝐴..^𝐵) ∪ (𝐵..^(𝐵 + 1)))) |
10 | fzosn 13649 | . . . 4 ⊢ (𝐵 ∈ ℤ → (𝐵..^(𝐵 + 1)) = {𝐵}) | |
11 | 2, 10 | syl 17 | . . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵..^(𝐵 + 1)) = {𝐵}) |
12 | 11 | uneq2d 4124 | . 2 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → ((𝐴..^𝐵) ∪ (𝐵..^(𝐵 + 1))) = ((𝐴..^𝐵) ∪ {𝐵})) |
13 | 9, 12 | eqtrd 2773 | 1 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴..^(𝐵 + 1)) = ((𝐴..^𝐵) ∪ {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∪ cun 3909 {csn 4587 ‘cfv 6497 (class class class)co 7358 1c1 11057 + caddc 11059 ℤcz 12504 ℤ≥cuz 12768 ...cfz 13430 ..^cfzo 13573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 |
This theorem is referenced by: fzosplitpr 13687 fzosplitsni 13689 fzisfzounsn 13690 cats1un 14615 bitsinv1 16327 bitsinvp1 16334 gsmsymgrfixlem1 19214 gsmsymgreqlem2 19218 efgsp1 19524 pgpfaclem1 19865 tgcgr4 27515 wlkp1lem8 28670 wlkp1 28671 crctcshwlkn0lem7 28803 clwlkclwwlklem2a1 28978 clwwlkel 29032 clwwlkwwlksb 29040 wwlksext2clwwlk 29043 eupthp1 29202 fzodif2 31742 fiunelros 32830 signsplypnf 33219 prodfzo03 33273 reprsuc 33285 breprexplema 33300 breprexplemc 33302 nnsum4primeseven 46078 nnsum4primesevenALTV 46079 |
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