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Mirrors > Home > ILE Home > Th. List > iseqfveq | GIF version |
Description: Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.) |
Ref | Expression |
---|---|
iseqfveq.1 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
iseqfveq.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
iseqfveq.f | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
iseqfveq.g | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) |
iseqfveq.pl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
Ref | Expression |
---|---|
iseqfveq | ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝑀( + , 𝐺, 𝑆)‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqfveq.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | eluzel2 8933 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
4 | uzid 8942 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
5 | 3, 4 | syl 14 | . 2 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
6 | iseqfveq.f | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) | |
7 | iseqfveq.pl | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
8 | 3, 6, 7 | iseq1 9766 | . . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹‘𝑀)) |
9 | fveq2 5256 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀)) | |
10 | fveq2 5256 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝐺‘𝑘) = (𝐺‘𝑀)) | |
11 | 9, 10 | eqeq12d 2099 | . . . 4 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘𝑀) = (𝐺‘𝑀))) |
12 | iseqfveq.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘)) | |
13 | 12 | ralrimiva 2442 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘)) |
14 | eluzfz1 9354 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
15 | 1, 14 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
16 | 11, 13, 15 | rspcdva 2719 | . . 3 ⊢ (𝜑 → (𝐹‘𝑀) = (𝐺‘𝑀)) |
17 | 8, 16 | eqtrd 2117 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐺‘𝑀)) |
18 | iseqfveq.g | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) | |
19 | fzp1ss 9394 | . . . . 5 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) | |
20 | 3, 19 | syl 14 | . . . 4 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
21 | 20 | sselda 3012 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ (𝑀...𝑁)) |
22 | 21, 12 | syldan 276 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
23 | 5, 17, 6, 18, 7, 1, 22 | iseqfveq2 9777 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝑀( + , 𝐺, 𝑆)‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1287 ∈ wcel 1436 ⊆ wss 2986 ‘cfv 4972 (class class class)co 5594 1c1 7272 + caddc 7274 ℤcz 8660 ℤ≥cuz 8928 ...cfz 9333 seqcseq 9754 |
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 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-coll 3922 ax-sep 3925 ax-nul 3933 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-iinf 4369 ax-cnex 7357 ax-resscn 7358 ax-1cn 7359 ax-1re 7360 ax-icn 7361 ax-addcl 7362 ax-addrcl 7363 ax-mulcl 7364 ax-addcom 7366 ax-addass 7368 ax-distr 7370 ax-i2m1 7371 ax-0lt1 7372 ax-0id 7374 ax-rnegex 7375 ax-cnre 7377 ax-pre-ltirr 7378 ax-pre-ltwlin 7379 ax-pre-lttrn 7380 ax-pre-ltadd 7382 |
This theorem depends on definitions: df-bi 115 df-3or 923 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-reu 2362 df-rab 2364 df-v 2616 df-sbc 2829 df-csb 2922 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-int 3666 df-iun 3709 df-br 3815 df-opab 3869 df-mpt 3870 df-tr 3905 df-id 4087 df-iord 4160 df-on 4162 df-ilim 4163 df-suc 4165 df-iom 4372 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-rn 4415 df-res 4416 df-ima 4417 df-iota 4937 df-fun 4974 df-fn 4975 df-f 4976 df-f1 4977 df-fo 4978 df-f1o 4979 df-fv 4980 df-riota 5550 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-1st 5849 df-2nd 5850 df-recs 6005 df-frec 6091 df-pnf 7445 df-mnf 7446 df-xr 7447 df-ltxr 7448 df-le 7449 df-sub 7576 df-neg 7577 df-inn 8335 df-n0 8584 df-z 8661 df-uz 8929 df-fz 9334 df-iseq 9755 |
This theorem is referenced by: iseqfeq 9780 |
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