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Theorem seqfeq3 10317
 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 2140 . . . 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 10266 . . 3 (𝜑 → seq𝑀( + , 𝐹):(ℤ𝑀)⟶𝑆)
65ffnd 5281 . 2 (𝜑 → seq𝑀( + , 𝐹) Fn (ℤ𝑀))
7 seqfeq3.id . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))
87, 4eqeltrrd 2218 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
91, 2, 3, 8seqf 10266 . . 3 (𝜑 → seq𝑀(𝑄, 𝐹):(ℤ𝑀)⟶𝑆)
109ffnd 5281 . 2 (𝜑 → seq𝑀(𝑄, 𝐹) Fn (ℤ𝑀))
115ffvelrnda 5563 . . . 4 ((𝜑𝑎 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹)‘𝑎) ∈ 𝑆)
12 fvi 5486 . . . 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 469 . . . 4 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
153adantlr 469 . . . 4 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
16 simpr 109 . . . 4 ((𝜑𝑎 ∈ (ℤ𝑀)) → 𝑎 ∈ (ℤ𝑀))
177adantlr 469 . . . . 5 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))
18 fvi 5486 . . . . . 6 ((𝑥 + 𝑦) ∈ 𝑆 → ( I ‘(𝑥 + 𝑦)) = (𝑥 + 𝑦))
1914, 18syl 14 . . . . 5 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → ( I ‘(𝑥 + 𝑦)) = (𝑥 + 𝑦))
20 fvi 5486 . . . . . . 7 (𝑥𝑆 → ( I ‘𝑥) = 𝑥)
2120ad2antrl 482 . . . . . 6 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → ( I ‘𝑥) = 𝑥)
22 fvi 5486 . . . . . . 7 (𝑦𝑆 → ( I ‘𝑦) = 𝑦)
2322ad2antll 483 . . . . . 6 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → ( I ‘𝑦) = 𝑦)
2421, 23oveq12d 5800 . . . . 5 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (( I ‘𝑥)𝑄( I ‘𝑦)) = (𝑥𝑄𝑦))
2517, 19, 243eqtr4d 2183 . . . 4 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → ( I ‘(𝑥 + 𝑦)) = (( I ‘𝑥)𝑄( I ‘𝑦)))
26 fvi 5486 . . . . 5 ((𝐹𝑥) ∈ 𝑆 → ( I ‘(𝐹𝑥)) = (𝐹𝑥))
2715, 26syl 14 . . . 4 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → ( I ‘(𝐹𝑥)) = (𝐹𝑥))
288adantlr 469 . . . 4 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
2914, 15, 16, 25, 27, 15, 28seq3homo 10315 . . 3 ((𝜑𝑎 ∈ (ℤ𝑀)) → ( I ‘(seq𝑀( + , 𝐹)‘𝑎)) = (seq𝑀(𝑄, 𝐹)‘𝑎))
3013, 29eqtr3d 2175 . 2 ((𝜑𝑎 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹)‘𝑎) = (seq𝑀(𝑄, 𝐹)‘𝑎))
316, 10, 30eqfnfvd 5529 1 (𝜑 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481   I cid 4218  ‘cfv 5131  (class class class)co 5782  ℤcz 9079  ℤ≥cuz 9351  seqcseq 10250 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7736  ax-resscn 7737  ax-1cn 7738  ax-1re 7739  ax-icn 7740  ax-addcl 7741  ax-addrcl 7742  ax-mulcl 7743  ax-addcom 7745  ax-addass 7747  ax-distr 7749  ax-i2m1 7750  ax-0lt1 7751  ax-0id 7753  ax-rnegex 7754  ax-cnre 7756  ax-pre-ltirr 7757  ax-pre-ltwlin 7758  ax-pre-lttrn 7759  ax-pre-ltadd 7761 This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7827  df-mnf 7828  df-xr 7829  df-ltxr 7830  df-le 7831  df-sub 7960  df-neg 7961  df-inn 8746  df-n0 9003  df-z 9080  df-uz 9352  df-seqfrec 10251 This theorem is referenced by: (None)
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