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Mirrors > Home > MPE Home > Th. List > seqfeq2 | Structured version Visualization version GIF version |
Description: Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.) |
Ref | Expression |
---|---|
seqfveq2.1 | ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀)) |
seqfveq2.2 | ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺‘𝐾)) |
seqfeq2.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
Ref | Expression |
---|---|
seqfeq2 | ⊢ (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝐾)) = seq𝐾( + , 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqfveq2.1 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀)) | |
2 | eluzel2 12629 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
3 | seqfn 13775 | . . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)) | |
4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)) |
5 | uzss 12647 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝐾) ⊆ (ℤ≥‘𝑀)) | |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ (ℤ≥‘𝑀)) |
7 | fnssres 6582 | . . 3 ⊢ ((seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀) ∧ (ℤ≥‘𝐾) ⊆ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝐾)) Fn (ℤ≥‘𝐾)) | |
8 | 4, 6, 7 | syl2anc 585 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝐾)) Fn (ℤ≥‘𝐾)) |
9 | eluzelz 12634 | . . 3 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ ℤ) | |
10 | seqfn 13775 | . . 3 ⊢ (𝐾 ∈ ℤ → seq𝐾( + , 𝐺) Fn (ℤ≥‘𝐾)) | |
11 | 1, 9, 10 | 3syl 18 | . 2 ⊢ (𝜑 → seq𝐾( + , 𝐺) Fn (ℤ≥‘𝐾)) |
12 | fvres 6819 | . . . 4 ⊢ (𝑥 ∈ (ℤ≥‘𝐾) → ((seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝐾))‘𝑥) = (seq𝑀( + , 𝐹)‘𝑥)) | |
13 | 12 | adantl 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → ((seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝐾))‘𝑥) = (seq𝑀( + , 𝐹)‘𝑥)) |
14 | 1 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → 𝐾 ∈ (ℤ≥‘𝑀)) |
15 | seqfveq2.2 | . . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺‘𝐾)) | |
16 | 15 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺‘𝐾)) |
17 | simpr 486 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → 𝑥 ∈ (ℤ≥‘𝐾)) | |
18 | elfzuz 13294 | . . . . . 6 ⊢ (𝑘 ∈ ((𝐾 + 1)...𝑥) → 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) | |
19 | seqfeq2.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) → (𝐹‘𝑘) = (𝐺‘𝑘)) | |
20 | 18, 19 | sylan2 594 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑥)) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
21 | 20 | adantlr 713 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑥)) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
22 | 14, 16, 17, 21 | seqfveq2 13787 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑥)) |
23 | 13, 22 | eqtrd 2776 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → ((seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝐾))‘𝑥) = (seq𝐾( + , 𝐺)‘𝑥)) |
24 | 8, 11, 23 | eqfnfvd 6940 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝐾)) = seq𝐾( + , 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1539 ∈ wcel 2104 ⊆ wss 3892 ↾ cres 5598 Fn wfn 6449 ‘cfv 6454 (class class class)co 7303 1c1 10914 + caddc 10916 ℤcz 12361 ℤ≥cuz 12624 ...cfz 13281 seqcseq 13763 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-om 7741 df-1st 7859 df-2nd 7860 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-er 8525 df-en 8761 df-dom 8762 df-sdom 8763 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-nn 12016 df-n0 12276 df-z 12362 df-uz 12625 df-fz 13282 df-seq 13764 |
This theorem is referenced by: seqid 13810 |
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