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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.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 | eqid 2082 | . . . 4 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
2 | iseqfeq.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | iseqfeq.f | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) | |
4 | iseqfeq.pl | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
5 | 1, 2, 3, 4 | iseqfcl 9535 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆):(ℤ≥‘𝑀)⟶𝑆) |
6 | ffn 5077 | . . 3 ⊢ (seq𝑀( + , 𝐹, 𝑆):(ℤ≥‘𝑀)⟶𝑆 → seq𝑀( + , 𝐹, 𝑆) Fn (ℤ≥‘𝑀)) | |
7 | 5, 6 | syl 14 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) Fn (ℤ≥‘𝑀)) |
8 | iseqfeq.2 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐺‘𝑘)) | |
9 | 8 | ralrimiva 2435 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) = (𝐺‘𝑘)) |
10 | fveq2 5209 | . . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥)) | |
11 | fveq2 5209 | . . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝐺‘𝑘) = (𝐺‘𝑥)) | |
12 | 10, 11 | eqeq12d 2096 | . . . . . . 7 ⊢ (𝑘 = 𝑥 → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘𝑥) = (𝐺‘𝑥))) |
13 | 12 | rspcv 2698 | . . . . . 6 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) = (𝐺‘𝑘) → (𝐹‘𝑥) = (𝐺‘𝑥))) |
14 | 9, 13 | mpan9 275 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
15 | 14, 3 | eqeltrrd 2157 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) |
16 | 1, 2, 15, 4 | iseqfcl 9535 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐺, 𝑆):(ℤ≥‘𝑀)⟶𝑆) |
17 | ffn 5077 | . . 3 ⊢ (seq𝑀( + , 𝐺, 𝑆):(ℤ≥‘𝑀)⟶𝑆 → seq𝑀( + , 𝐺, 𝑆) Fn (ℤ≥‘𝑀)) | |
18 | 16, 17 | syl 14 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐺, 𝑆) Fn (ℤ≥‘𝑀)) |
19 | simpr 108 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → 𝑧 ∈ (ℤ≥‘𝑀)) | |
20 | elfzuz 9117 | . . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑧) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
21 | 20, 8 | sylan2 280 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑧)) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
22 | 21 | adantlr 461 | . . 3 ⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...𝑧)) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
23 | 3 | adantlr 461 | . . 3 ⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
24 | 15 | adantlr 461 | . . 3 ⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) |
25 | 4 | adantlr 461 | . . 3 ⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
26 | 19, 22, 23, 24, 25 | iseqfveq 9546 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐺, 𝑆)‘𝑧)) |
27 | 7, 18, 26 | eqfnfvd 5300 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 ∀wral 2349 Fn wfn 4927 ⟶wf 4928 ‘cfv 4932 (class class class)co 5543 ℤcz 8432 ℤ≥cuz 8700 ...cfz 9105 seqcseq 9521 |
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 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-coll 3901 ax-sep 3904 ax-nul 3912 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-iinf 4337 ax-cnex 7129 ax-resscn 7130 ax-1cn 7131 ax-1re 7132 ax-icn 7133 ax-addcl 7134 ax-addrcl 7135 ax-mulcl 7136 ax-addcom 7138 ax-addass 7140 ax-distr 7142 ax-i2m1 7143 ax-0lt1 7144 ax-0id 7146 ax-rnegex 7147 ax-cnre 7149 ax-pre-ltirr 7150 ax-pre-ltwlin 7151 ax-pre-lttrn 7152 ax-pre-ltadd 7154 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-nel 2341 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-csb 2910 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-nul 3259 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-int 3645 df-iun 3688 df-br 3794 df-opab 3848 df-mpt 3849 df-tr 3884 df-id 4056 df-iord 4129 df-on 4131 df-ilim 4132 df-suc 4134 df-iom 4340 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-rn 4382 df-res 4383 df-ima 4384 df-iota 4897 df-fun 4934 df-fn 4935 df-f 4936 df-f1 4937 df-fo 4938 df-f1o 4939 df-fv 4940 df-riota 5499 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-1st 5798 df-2nd 5799 df-recs 5954 df-frec 6040 df-pnf 7217 df-mnf 7218 df-xr 7219 df-ltxr 7220 df-le 7221 df-sub 7348 df-neg 7349 df-inn 8107 df-n0 8356 df-z 8433 df-uz 8701 df-fz 9106 df-iseq 9522 |
This theorem is referenced by: (None) |
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