![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > seq3fveq | 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 |
---|---|
seq3fveq | ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqfveq.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | eluzel2 9551 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
4 | uzid 9560 | . . 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 | seq3-1 10478 | . . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
9 | fveq2 5530 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀)) | |
10 | fveq2 5530 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝐺‘𝑘) = (𝐺‘𝑀)) | |
11 | 9, 10 | eqeq12d 2204 | . . . 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 2563 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘)) |
14 | eluzfz1 10049 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
15 | 1, 14 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
16 | 11, 13, 15 | rspcdva 2861 | . . 3 ⊢ (𝜑 → (𝐹‘𝑀) = (𝐺‘𝑀)) |
17 | 8, 16 | eqtrd 2222 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐺‘𝑀)) |
18 | iseqfveq.g | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) | |
19 | fzp1ss 10091 | . . . . 5 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) | |
20 | 3, 19 | syl 14 | . . . 4 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
21 | 20 | sselda 3170 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ (𝑀...𝑁)) |
22 | 21, 12 | syldan 282 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
23 | 5, 17, 6, 18, 7, 1, 22 | seq3fveq2 10487 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 ⊆ wss 3144 ‘cfv 5231 (class class class)co 5891 1c1 7830 + caddc 7832 ℤcz 9271 ℤ≥cuz 9546 ...cfz 10026 seqcseq 10463 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-addcom 7929 ax-addass 7931 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-0id 7937 ax-rnegex 7938 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-ltadd 7945 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-frec 6410 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-inn 8938 df-n0 9195 df-z 9272 df-uz 9547 df-fz 10027 df-seqfrec 10464 |
This theorem is referenced by: seq3feq 10490 seq3f1olemqsumk 10517 seq3f1olemqsum 10518 seq3f1oleml 10521 seq3f1o 10522 fsum3 11413 fsum3ser 11423 fprodseq 11609 fprodntrivap 11610 |
Copyright terms: Public domain | W3C validator |