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Mirrors > Home > ILE Home > Th. List > iseqfeq | GIF version |
Description: Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) |
Ref | Expression |
---|---|
iseqfeq.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
iseqfeq.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
iseqfeq.f | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
iseqfeq.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
iseqfeq.pl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
Ref | Expression |
---|---|
iseqfeq | ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqfeq.1 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | iseqfeq.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
3 | iseqfeq.f | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) | |
4 | iseqfeq.pl | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
5 | 1, 2, 3, 4 | iseqfn 9221 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) Fn (ℤ≥‘𝑀)) |
6 | iseqfeq.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐺‘𝑘)) | |
7 | 6 | ralrimiva 2392 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) = (𝐺‘𝑘)) |
8 | fveq2 5178 | . . . . . . 7 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥)) | |
9 | fveq2 5178 | . . . . . . 7 ⊢ (𝑘 = 𝑥 → (𝐺‘𝑘) = (𝐺‘𝑥)) | |
10 | 8, 9 | eqeq12d 2054 | . . . . . 6 ⊢ (𝑘 = 𝑥 → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘𝑥) = (𝐺‘𝑥))) |
11 | 10 | rspcv 2652 | . . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) = (𝐺‘𝑘) → (𝐹‘𝑥) = (𝐺‘𝑥))) |
12 | 7, 11 | mpan9 265 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
13 | 12, 3 | eqeltrrd 2115 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) |
14 | 1, 2, 13, 4 | iseqfn 9221 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐺, 𝑆) Fn (ℤ≥‘𝑀)) |
15 | simpr 103 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → 𝑧 ∈ (ℤ≥‘𝑀)) | |
16 | elfzuz 8886 | . . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑧) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
17 | 16, 6 | sylan2 270 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑧)) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
18 | 17 | adantlr 446 | . . 3 ⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...𝑧)) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
19 | 2 | adantr 261 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → 𝑆 ∈ 𝑉) |
20 | 3 | adantlr 446 | . . 3 ⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
21 | 13 | adantlr 446 | . . 3 ⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) |
22 | 4 | adantlr 446 | . . 3 ⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
23 | 15, 18, 19, 20, 21, 22 | iseqfveq 9230 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐺, 𝑆)‘𝑧)) |
24 | 5, 14, 23 | eqfnfvd 5268 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ∀wral 2306 ‘cfv 4902 (class class class)co 5512 ℤcz 8245 ℤ≥cuz 8473 ...cfz 8874 seqcseq 9211 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-ltadd 7000 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-frec 5978 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-inn 7915 df-n0 8182 df-z 8246 df-uz 8474 df-fz 8875 df-iseq 9212 |
This theorem is referenced by: (None) |
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