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| Mirrors > Home > MPE Home > Th. List > fzsuc2 | Structured version Visualization version GIF version | ||
| Description: Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.) | 
| Ref | Expression | 
|---|---|
| fzsuc2 | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 − 1))) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | uzp1 12919 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 − 1)) → (𝑁 = (𝑀 − 1) ∨ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1)))) | |
| 2 | zcn 12618 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
| 3 | ax-1cn 11213 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
| 4 | npcan 11517 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 − 1) + 1) = 𝑀) | |
| 5 | 2, 3, 4 | sylancl 586 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 1) + 1) = 𝑀) | 
| 6 | 5 | oveq2d 7447 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝑀...((𝑀 − 1) + 1)) = (𝑀...𝑀)) | 
| 7 | uncom 4158 | . . . . . . . 8 ⊢ (∅ ∪ {𝑀}) = ({𝑀} ∪ ∅) | |
| 8 | un0 4394 | . . . . . . . 8 ⊢ ({𝑀} ∪ ∅) = {𝑀} | |
| 9 | 7, 8 | eqtri 2765 | . . . . . . 7 ⊢ (∅ ∪ {𝑀}) = {𝑀} | 
| 10 | zre 12617 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
| 11 | 10 | ltm1d 12200 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) < 𝑀) | 
| 12 | peano2zm 12660 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ) | |
| 13 | fzn 13580 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅)) | |
| 14 | 12, 13 | mpdan 687 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅)) | 
| 15 | 11, 14 | mpbid 232 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 − 1)) = ∅) | 
| 16 | 5 | sneqd 4638 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → {((𝑀 − 1) + 1)} = {𝑀}) | 
| 17 | 15, 16 | uneq12d 4169 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)}) = (∅ ∪ {𝑀})) | 
| 18 | fzsn 13606 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) | |
| 19 | 9, 17, 18 | 3eqtr4a 2803 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)}) = (𝑀...𝑀)) | 
| 20 | 6, 19 | eqtr4d 2780 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀...((𝑀 − 1) + 1)) = ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)})) | 
| 21 | oveq1 7438 | . . . . . . 7 ⊢ (𝑁 = (𝑀 − 1) → (𝑁 + 1) = ((𝑀 − 1) + 1)) | |
| 22 | 21 | oveq2d 7447 | . . . . . 6 ⊢ (𝑁 = (𝑀 − 1) → (𝑀...(𝑁 + 1)) = (𝑀...((𝑀 − 1) + 1))) | 
| 23 | oveq2 7439 | . . . . . . 7 ⊢ (𝑁 = (𝑀 − 1) → (𝑀...𝑁) = (𝑀...(𝑀 − 1))) | |
| 24 | 21 | sneqd 4638 | . . . . . . 7 ⊢ (𝑁 = (𝑀 − 1) → {(𝑁 + 1)} = {((𝑀 − 1) + 1)}) | 
| 25 | 23, 24 | uneq12d 4169 | . . . . . 6 ⊢ (𝑁 = (𝑀 − 1) → ((𝑀...𝑁) ∪ {(𝑁 + 1)}) = ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)})) | 
| 26 | 22, 25 | eqeq12d 2753 | . . . . 5 ⊢ (𝑁 = (𝑀 − 1) → ((𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}) ↔ (𝑀...((𝑀 − 1) + 1)) = ((𝑀...(𝑀 − 1)) ∪ {((𝑀 − 1) + 1)}))) | 
| 27 | 20, 26 | syl5ibrcom 247 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑁 = (𝑀 − 1) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}))) | 
| 28 | 27 | imp 406 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 = (𝑀 − 1)) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) | 
| 29 | 5 | fveq2d 6910 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘((𝑀 − 1) + 1)) = (ℤ≥‘𝑀)) | 
| 30 | 29 | eleq2d 2827 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1)) ↔ 𝑁 ∈ (ℤ≥‘𝑀))) | 
| 31 | 30 | biimpa 476 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1))) → 𝑁 ∈ (ℤ≥‘𝑀)) | 
| 32 | fzsuc 13611 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) | |
| 33 | 31, 32 | syl 17 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1))) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) | 
| 34 | 28, 33 | jaodan 960 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 = (𝑀 − 1) ∨ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1)))) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) | 
| 35 | 1, 34 | sylan2 593 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 − 1))) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ∅c0 4333 {csn 4626 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 < clt 11295 − cmin 11492 ℤcz 12613 ℤ≥cuz 12878 ...cfz 13547 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 | 
| This theorem is referenced by: fseq1p1m1 13638 fzennn 14009 fsumm1 15787 fprodm1 16003 prmreclem4 16957 ppiprm 27194 ppinprm 27195 chtprm 27196 chtnprm 27197 poimirlem3 37630 poimirlem4 37631 lcmfunnnd 42013 mapfzcons 42727 | 
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