![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > seqfeq3 | Structured version Visualization version 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 | seqfeq3.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | seqfn 13848 | . . 3 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)) |
4 | seqfn 13848 | . . 3 ⊢ (𝑀 ∈ ℤ → seq𝑀(𝑄, 𝐹) Fn (ℤ≥‘𝑀)) | |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝜑 → seq𝑀(𝑄, 𝐹) Fn (ℤ≥‘𝑀)) |
6 | simpr 486 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → 𝑎 ∈ (ℤ≥‘𝑀)) | |
7 | simpll 766 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (𝑀...𝑎)) → 𝜑) | |
8 | elfzuz 13367 | . . . . 5 ⊢ (𝑥 ∈ (𝑀...𝑎) → 𝑥 ∈ (ℤ≥‘𝑀)) | |
9 | 8 | adantl 483 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (𝑀...𝑎)) → 𝑥 ∈ (ℤ≥‘𝑀)) |
10 | seqfeq3.f | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) | |
11 | 7, 9, 10 | syl2anc 585 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (𝑀...𝑎)) → (𝐹‘𝑥) ∈ 𝑆) |
12 | seqfeq3.cl | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
13 | 12 | adantlr 714 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
14 | seqfeq3.id | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦)) | |
15 | 14 | adantlr 714 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦)) |
16 | 6, 11, 13, 15 | seqfeq4 13887 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹)‘𝑎) = (seq𝑀(𝑄, 𝐹)‘𝑎)) |
17 | 3, 5, 16 | eqfnfvd 6981 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Fn wfn 6487 ‘cfv 6492 (class class class)co 7350 ℤcz 12433 ℤ≥cuz 12697 ...cfz 13354 seqcseq 13836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-n0 12348 df-z 12434 df-uz 12698 df-fz 13355 df-seq 13837 |
This theorem is referenced by: mulgpropd 18853 esumfsupre 32450 |
Copyright terms: Public domain | W3C validator |