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Theorem seqfeq3 10172
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.)
Hypotheses
Ref Expression
seqfeq3.m (𝜑𝑀 ∈ ℤ)
seqfeq3.f ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
seqfeq3.cl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seqfeq3.id ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))
Assertion
Ref Expression
seqfeq3 (𝜑 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦   𝑥,𝑀,𝑦   𝑥, + ,𝑦   𝑥,𝑄,𝑦   𝑥,𝑆,𝑦

Proof of Theorem seqfeq3
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eqid 2113 . . . 4 (ℤ𝑀) = (ℤ𝑀)
2 seqfeq3.m . . . 4 (𝜑𝑀 ∈ ℤ)
3 seqfeq3.f . . . 4 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
4 seqfeq3.cl . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
51, 2, 3, 4seqf 10121 . . 3 (𝜑 → seq𝑀( + , 𝐹):(ℤ𝑀)⟶𝑆)
65ffnd 5229 . 2 (𝜑 → seq𝑀( + , 𝐹) Fn (ℤ𝑀))
7 seqfeq3.id . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))
87, 4eqeltrrd 2190 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
91, 2, 3, 8seqf 10121 . . 3 (𝜑 → seq𝑀(𝑄, 𝐹):(ℤ𝑀)⟶𝑆)
109ffnd 5229 . 2 (𝜑 → seq𝑀(𝑄, 𝐹) Fn (ℤ𝑀))
115ffvelrnda 5507 . . . 4 ((𝜑𝑎 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹)‘𝑎) ∈ 𝑆)
12 fvi 5430 . . . 4 ((seq𝑀( + , 𝐹)‘𝑎) ∈ 𝑆 → ( I ‘(seq𝑀( + , 𝐹)‘𝑎)) = (seq𝑀( + , 𝐹)‘𝑎))
1311, 12syl 14 . . 3 ((𝜑𝑎 ∈ (ℤ𝑀)) → ( I ‘(seq𝑀( + , 𝐹)‘𝑎)) = (seq𝑀( + , 𝐹)‘𝑎))
144adantlr 466 . . . 4 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
153adantlr 466 . . . 4 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
16 simpr 109 . . . 4 ((𝜑𝑎 ∈ (ℤ𝑀)) → 𝑎 ∈ (ℤ𝑀))
177adantlr 466 . . . . 5 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))
18 fvi 5430 . . . . . 6 ((𝑥 + 𝑦) ∈ 𝑆 → ( I ‘(𝑥 + 𝑦)) = (𝑥 + 𝑦))
1914, 18syl 14 . . . . 5 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → ( I ‘(𝑥 + 𝑦)) = (𝑥 + 𝑦))
20 fvi 5430 . . . . . . 7 (𝑥𝑆 → ( I ‘𝑥) = 𝑥)
2120ad2antrl 479 . . . . . 6 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → ( I ‘𝑥) = 𝑥)
22 fvi 5430 . . . . . . 7 (𝑦𝑆 → ( I ‘𝑦) = 𝑦)
2322ad2antll 480 . . . . . 6 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → ( I ‘𝑦) = 𝑦)
2421, 23oveq12d 5744 . . . . 5 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (( I ‘𝑥)𝑄( I ‘𝑦)) = (𝑥𝑄𝑦))
2517, 19, 243eqtr4d 2155 . . . 4 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → ( I ‘(𝑥 + 𝑦)) = (( I ‘𝑥)𝑄( I ‘𝑦)))
26 fvi 5430 . . . . 5 ((𝐹𝑥) ∈ 𝑆 → ( I ‘(𝐹𝑥)) = (𝐹𝑥))
2715, 26syl 14 . . . 4 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → ( I ‘(𝐹𝑥)) = (𝐹𝑥))
288adantlr 466 . . . 4 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
2914, 15, 16, 25, 27, 15, 28seq3homo 10170 . . 3 ((𝜑𝑎 ∈ (ℤ𝑀)) → ( I ‘(seq𝑀( + , 𝐹)‘𝑎)) = (seq𝑀(𝑄, 𝐹)‘𝑎))
3013, 29eqtr3d 2147 . 2 ((𝜑𝑎 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹)‘𝑎) = (seq𝑀(𝑄, 𝐹)‘𝑎))
316, 10, 30eqfnfvd 5473 1 (𝜑 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1312  wcel 1461   I cid 4168  cfv 5079  (class class class)co 5726  cz 8952  cuz 9222  seqcseq 10105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-addcom 7639  ax-addass 7641  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-0id 7647  ax-rnegex 7648  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-ltadd 7655
This theorem depends on definitions:  df-bi 116  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-ilim 4249  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-frec 6240  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-inn 8625  df-n0 8876  df-z 8953  df-uz 9223  df-seqfrec 10106
This theorem is referenced by: (None)
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