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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 12813 | . . 3 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)) |
4 | seqfn 12813 | . . 3 ⊢ (𝑀 ∈ ℤ → seq𝑀(𝑄, 𝐹) Fn (ℤ≥‘𝑀)) | |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝜑 → seq𝑀(𝑄, 𝐹) Fn (ℤ≥‘𝑀)) |
6 | simpr 477 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → 𝑎 ∈ (ℤ≥‘𝑀)) | |
7 | simpll 790 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (𝑀...𝑎)) → 𝜑) | |
8 | elfzuz 12338 | . . . . 5 ⊢ (𝑥 ∈ (𝑀...𝑎) → 𝑥 ∈ (ℤ≥‘𝑀)) | |
9 | 8 | adantl 482 | . . . 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 693 | . . 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 751 | . . 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 751 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦)) |
16 | 6, 11, 13, 15 | seqfeq4 12850 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹)‘𝑎) = (seq𝑀(𝑄, 𝐹)‘𝑎)) |
17 | 3, 5, 16 | eqfnfvd 6314 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 ℤcz 11377 ℤ≥cuz 11687 ...cfz 12326 seqcseq 12801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-seq 12802 |
This theorem is referenced by: mulgpropd 17584 esumfsupre 30133 |
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