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Mirrors > Home > ILE Home > Th. List > seqfeq3 | GIF version |
Description: Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
seqfeq3.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
seqfeq3.f | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
seqfeq3.cl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
seqfeq3.id | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦)) |
Ref | Expression |
---|---|
seqfeq3 | ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2137 | . . . 4 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
2 | seqfeq3.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | seqfeq3.f | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) | |
4 | seqfeq3.cl | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
5 | 1, 2, 3, 4 | seqf 10227 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶𝑆) |
6 | 5 | ffnd 5268 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)) |
7 | seqfeq3.id | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦)) | |
8 | 7, 4 | eqeltrrd 2215 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆) |
9 | 1, 2, 3, 8 | seqf 10227 | . . 3 ⊢ (𝜑 → seq𝑀(𝑄, 𝐹):(ℤ≥‘𝑀)⟶𝑆) |
10 | 9 | ffnd 5268 | . 2 ⊢ (𝜑 → seq𝑀(𝑄, 𝐹) Fn (ℤ≥‘𝑀)) |
11 | 5 | ffvelrnda 5548 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹)‘𝑎) ∈ 𝑆) |
12 | fvi 5471 | . . . 4 ⊢ ((seq𝑀( + , 𝐹)‘𝑎) ∈ 𝑆 → ( I ‘(seq𝑀( + , 𝐹)‘𝑎)) = (seq𝑀( + , 𝐹)‘𝑎)) | |
13 | 11, 12 | syl 14 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → ( I ‘(seq𝑀( + , 𝐹)‘𝑎)) = (seq𝑀( + , 𝐹)‘𝑎)) |
14 | 4 | adantlr 468 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
15 | 3 | adantlr 468 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
16 | simpr 109 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → 𝑎 ∈ (ℤ≥‘𝑀)) | |
17 | 7 | adantlr 468 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦)) |
18 | fvi 5471 | . . . . . 6 ⊢ ((𝑥 + 𝑦) ∈ 𝑆 → ( I ‘(𝑥 + 𝑦)) = (𝑥 + 𝑦)) | |
19 | 14, 18 | syl 14 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ( I ‘(𝑥 + 𝑦)) = (𝑥 + 𝑦)) |
20 | fvi 5471 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑆 → ( I ‘𝑥) = 𝑥) | |
21 | 20 | ad2antrl 481 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ( I ‘𝑥) = 𝑥) |
22 | fvi 5471 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑆 → ( I ‘𝑦) = 𝑦) | |
23 | 22 | ad2antll 482 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ( I ‘𝑦) = 𝑦) |
24 | 21, 23 | oveq12d 5785 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (( I ‘𝑥)𝑄( I ‘𝑦)) = (𝑥𝑄𝑦)) |
25 | 17, 19, 24 | 3eqtr4d 2180 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ( I ‘(𝑥 + 𝑦)) = (( I ‘𝑥)𝑄( I ‘𝑦))) |
26 | fvi 5471 | . . . . 5 ⊢ ((𝐹‘𝑥) ∈ 𝑆 → ( I ‘(𝐹‘𝑥)) = (𝐹‘𝑥)) | |
27 | 15, 26 | syl 14 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ( I ‘(𝐹‘𝑥)) = (𝐹‘𝑥)) |
28 | 8 | adantlr 468 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆) |
29 | 14, 15, 16, 25, 27, 15, 28 | seq3homo 10276 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → ( I ‘(seq𝑀( + , 𝐹)‘𝑎)) = (seq𝑀(𝑄, 𝐹)‘𝑎)) |
30 | 13, 29 | eqtr3d 2172 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹)‘𝑎) = (seq𝑀(𝑄, 𝐹)‘𝑎)) |
31 | 6, 10, 30 | eqfnfvd 5514 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 I cid 4205 ‘cfv 5118 (class class class)co 5767 ℤcz 9047 ℤ≥cuz 9319 seqcseq 10211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-seqfrec 10212 |
This theorem is referenced by: (None) |
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