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Mirrors > Home > ILE Home > Th. List > isumrb | GIF version |
Description: Rebase the starting point of a sum. (Contributed by Jim Kingdon, 5-Mar-2022.) |
Ref | Expression |
---|---|
isummo.1 | ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) |
isummo.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
isumrb.4 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
isumrb.5 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
isumrb.6 | ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀)) |
isumrb.7 | ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑁)) |
isumrb.mdc | ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴) |
isumrb.ndc | ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → DECID 𝑘 ∈ 𝐴) |
Ref | Expression |
---|---|
isumrb | ⊢ (𝜑 → (seq𝑀( + , 𝐹, ℂ) ⇝ 𝐶 ↔ seq𝑁( + , 𝐹, ℂ) ⇝ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumrb.5 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
2 | 1 | adantr 270 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℤ) |
3 | iseqex 9742 | . . . 4 ⊢ seq𝑀( + , 𝐹, ℂ) ∈ V | |
4 | climres 10516 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ seq𝑀( + , 𝐹, ℂ) ∈ V) → ((seq𝑀( + , 𝐹, ℂ) ↾ (ℤ≥‘𝑁)) ⇝ 𝐶 ↔ seq𝑀( + , 𝐹, ℂ) ⇝ 𝐶)) | |
5 | 2, 3, 4 | sylancl 404 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ((seq𝑀( + , 𝐹, ℂ) ↾ (ℤ≥‘𝑁)) ⇝ 𝐶 ↔ seq𝑀( + , 𝐹, ℂ) ⇝ 𝐶)) |
6 | isumrb.7 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑁)) | |
7 | isummo.1 | . . . . . 6 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) | |
8 | isummo.2 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
9 | 8 | adantlr 461 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
10 | isumrb.mdc | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴) | |
11 | 10 | adantlr 461 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴) |
12 | simpr 108 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ (ℤ≥‘𝑀)) | |
13 | 7, 9, 11, 12 | isumrblem 10573 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) → (seq𝑀( + , 𝐹, ℂ) ↾ (ℤ≥‘𝑁)) = seq𝑁( + , 𝐹, ℂ)) |
14 | 6, 13 | mpidan 414 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹, ℂ) ↾ (ℤ≥‘𝑁)) = seq𝑁( + , 𝐹, ℂ)) |
15 | 14 | breq1d 3821 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ((seq𝑀( + , 𝐹, ℂ) ↾ (ℤ≥‘𝑁)) ⇝ 𝐶 ↔ seq𝑁( + , 𝐹, ℂ) ⇝ 𝐶)) |
16 | 5, 15 | bitr3d 188 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹, ℂ) ⇝ 𝐶 ↔ seq𝑁( + , 𝐹, ℂ) ⇝ 𝐶)) |
17 | isumrb.6 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀)) | |
18 | 8 | adantlr 461 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
19 | isumrb.ndc | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → DECID 𝑘 ∈ 𝐴) | |
20 | 19 | adantlr 461 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → DECID 𝑘 ∈ 𝐴) |
21 | simpr 108 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) | |
22 | 7, 18, 20, 21 | isumrblem 10573 | . . . . 5 ⊢ (((𝜑 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) ∧ 𝐴 ⊆ (ℤ≥‘𝑀)) → (seq𝑁( + , 𝐹, ℂ) ↾ (ℤ≥‘𝑀)) = seq𝑀( + , 𝐹, ℂ)) |
23 | 17, 22 | mpidan 414 | . . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → (seq𝑁( + , 𝐹, ℂ) ↾ (ℤ≥‘𝑀)) = seq𝑀( + , 𝐹, ℂ)) |
24 | 23 | breq1d 3821 | . . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → ((seq𝑁( + , 𝐹, ℂ) ↾ (ℤ≥‘𝑀)) ⇝ 𝐶 ↔ seq𝑀( + , 𝐹, ℂ) ⇝ 𝐶)) |
25 | isumrb.4 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
26 | 25 | adantr 270 | . . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℤ) |
27 | iseqex 9742 | . . . 4 ⊢ seq𝑁( + , 𝐹, ℂ) ∈ V | |
28 | climres 10516 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ seq𝑁( + , 𝐹, ℂ) ∈ V) → ((seq𝑁( + , 𝐹, ℂ) ↾ (ℤ≥‘𝑀)) ⇝ 𝐶 ↔ seq𝑁( + , 𝐹, ℂ) ⇝ 𝐶)) | |
29 | 26, 27, 28 | sylancl 404 | . . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → ((seq𝑁( + , 𝐹, ℂ) ↾ (ℤ≥‘𝑀)) ⇝ 𝐶 ↔ seq𝑁( + , 𝐹, ℂ) ⇝ 𝐶)) |
30 | 24, 29 | bitr3d 188 | . 2 ⊢ ((𝜑 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → (seq𝑀( + , 𝐹, ℂ) ⇝ 𝐶 ↔ seq𝑁( + , 𝐹, ℂ) ⇝ 𝐶)) |
31 | uztric 8935 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ∨ 𝑀 ∈ (ℤ≥‘𝑁))) | |
32 | 25, 1, 31 | syl2anc 403 | . 2 ⊢ (𝜑 → (𝑁 ∈ (ℤ≥‘𝑀) ∨ 𝑀 ∈ (ℤ≥‘𝑁))) |
33 | 16, 30, 32 | mpjaodan 745 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹, ℂ) ⇝ 𝐶 ↔ seq𝑁( + , 𝐹, ℂ) ⇝ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 662 DECID wdc 776 = wceq 1285 ∈ wcel 1434 Vcvv 2612 ⊆ wss 2984 ifcif 3373 class class class wbr 3811 ↦ cmpt 3865 ↾ cres 4403 ‘cfv 4969 ℂcc 7251 0cc0 7253 + caddc 7256 ℤcz 8646 ℤ≥cuz 8914 seqcseq 9740 ⇝ cli 10491 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-iinf 4366 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-addcom 7348 ax-addass 7350 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-0id 7356 ax-rnegex 7357 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-apti 7363 ax-pre-ltadd 7364 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-if 3374 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4084 df-iord 4157 df-on 4159 df-ilim 4160 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-1st 5846 df-2nd 5847 df-recs 6002 df-frec 6088 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-inn 8317 df-n0 8566 df-z 8647 df-uz 8915 df-fz 9320 df-fzo 9444 df-iseq 9741 df-clim 10492 |
This theorem is referenced by: (None) |
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