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Mirrors > Home > ILE Home > Th. List > seq3feq2 | GIF version |
Description: Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.) |
Ref | Expression |
---|---|
seq3fveq2.1 | ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀)) |
seq3fveq2.2 | ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺‘𝐾)) |
seq3fveq2.f | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
seq3fveq2.g | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑥) ∈ 𝑆) |
seq3fveq2.pl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
seq3feq2.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
Ref | Expression |
---|---|
seq3feq2 | ⊢ (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝐾)) = seq𝐾( + , 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . . . . 5 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
2 | seq3fveq2.1 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀)) | |
3 | eluzel2 9533 | . . . . . 6 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
4 | 2, 3 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
5 | seq3fveq2.f | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) | |
6 | seq3fveq2.pl | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
7 | 1, 4, 5, 6 | seqf 10461 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶𝑆) |
8 | 7 | ffnd 5367 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)) |
9 | uzss 9548 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝐾) ⊆ (ℤ≥‘𝑀)) | |
10 | 2, 9 | syl 14 | . . 3 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ (ℤ≥‘𝑀)) |
11 | fnssres 5330 | . . 3 ⊢ ((seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀) ∧ (ℤ≥‘𝐾) ⊆ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝐾)) Fn (ℤ≥‘𝐾)) | |
12 | 8, 10, 11 | syl2anc 411 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝐾)) Fn (ℤ≥‘𝐾)) |
13 | eqid 2177 | . . . 4 ⊢ (ℤ≥‘𝐾) = (ℤ≥‘𝐾) | |
14 | eluzelz 9537 | . . . . 5 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ ℤ) | |
15 | 2, 14 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
16 | seq3fveq2.g | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑥) ∈ 𝑆) | |
17 | 13, 15, 16, 6 | seqf 10461 | . . 3 ⊢ (𝜑 → seq𝐾( + , 𝐺):(ℤ≥‘𝐾)⟶𝑆) |
18 | 17 | ffnd 5367 | . 2 ⊢ (𝜑 → seq𝐾( + , 𝐺) Fn (ℤ≥‘𝐾)) |
19 | fvres 5540 | . . . 4 ⊢ (𝑧 ∈ (ℤ≥‘𝐾) → ((seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝐾))‘𝑧) = (seq𝑀( + , 𝐹)‘𝑧)) | |
20 | 19 | adantl 277 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝐾)) → ((seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝐾))‘𝑧) = (seq𝑀( + , 𝐹)‘𝑧)) |
21 | 2 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝐾)) → 𝐾 ∈ (ℤ≥‘𝑀)) |
22 | seq3fveq2.2 | . . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺‘𝐾)) | |
23 | 22 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝐾)) → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺‘𝐾)) |
24 | 5 | adantlr 477 | . . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝐾)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
25 | 16 | adantlr 477 | . . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝐾)) ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑥) ∈ 𝑆) |
26 | 6 | adantlr 477 | . . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝐾)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
27 | simpr 110 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝐾)) → 𝑧 ∈ (ℤ≥‘𝐾)) | |
28 | elfzuz 10021 | . . . . . 6 ⊢ (𝑘 ∈ ((𝐾 + 1)...𝑧) → 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) | |
29 | seq3feq2.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) → (𝐹‘𝑘) = (𝐺‘𝑘)) | |
30 | 28, 29 | sylan2 286 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑧)) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
31 | 30 | adantlr 477 | . . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝐾)) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑧)) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
32 | 21, 23, 24, 25, 26, 27, 31 | seq3fveq2 10469 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝐾)) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧)) |
33 | 20, 32 | eqtrd 2210 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝐾)) → ((seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝐾))‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧)) |
34 | 12, 18, 33 | eqfnfvd 5617 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝐾)) = seq𝐾( + , 𝐺)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ⊆ wss 3130 ↾ cres 4629 Fn wfn 5212 ‘cfv 5217 (class class class)co 5875 1c1 7812 + caddc 7814 ℤcz 9253 ℤ≥cuz 9528 ...cfz 10008 seqcseq 10445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-addcom 7911 ax-addass 7913 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-0id 7919 ax-rnegex 7920 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-ltadd 7927 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-frec 6392 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-inn 8920 df-n0 9177 df-z 9254 df-uz 9529 df-fz 10009 df-seqfrec 10446 |
This theorem is referenced by: seq3id 10508 |
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