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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 9546 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
4 | uzid 9555 | . . 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 10473 | . . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
9 | fveq2 5527 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀)) | |
10 | fveq2 5527 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝐺‘𝑘) = (𝐺‘𝑀)) | |
11 | 9, 10 | eqeq12d 2202 | . . . 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 2560 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘)) |
14 | eluzfz1 10044 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
15 | 1, 14 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
16 | 11, 13, 15 | rspcdva 2858 | . . 3 ⊢ (𝜑 → (𝐹‘𝑀) = (𝐺‘𝑀)) |
17 | 8, 16 | eqtrd 2220 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐺‘𝑀)) |
18 | iseqfveq.g | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) | |
19 | fzp1ss 10086 | . . . . 5 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) | |
20 | 3, 19 | syl 14 | . . . 4 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
21 | 20 | sselda 3167 | . . 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 10482 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2158 ⊆ wss 3141 ‘cfv 5228 (class class class)co 5888 1c1 7825 + caddc 7827 ℤcz 9266 ℤ≥cuz 9541 ...cfz 10021 seqcseq 10458 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-frec 6405 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-inn 8933 df-n0 9190 df-z 9267 df-uz 9542 df-fz 10022 df-seqfrec 10459 |
This theorem is referenced by: seq3feq 10485 seq3f1olemqsumk 10512 seq3f1olemqsum 10513 seq3f1oleml 10516 seq3f1o 10517 fsum3 11408 fsum3ser 11418 fprodseq 11604 fprodntrivap 11605 |
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