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Theorem seqfeq3 10545
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 2189 . . . 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 10494 . . 3 (𝜑 → seq𝑀( + , 𝐹):(ℤ𝑀)⟶𝑆)
65ffnd 5385 . 2 (𝜑 → seq𝑀( + , 𝐹) Fn (ℤ𝑀))
7 seqfeq3.id . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))
87, 4eqeltrrd 2267 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
91, 2, 3, 8seqf 10494 . . 3 (𝜑 → seq𝑀(𝑄, 𝐹):(ℤ𝑀)⟶𝑆)
109ffnd 5385 . 2 (𝜑 → seq𝑀(𝑄, 𝐹) Fn (ℤ𝑀))
115ffvelcdmda 5672 . . . 4 ((𝜑𝑎 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹)‘𝑎) ∈ 𝑆)
12 fvi 5594 . . . 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 477 . . . 4 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
153adantlr 477 . . . 4 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
16 simpr 110 . . . 4 ((𝜑𝑎 ∈ (ℤ𝑀)) → 𝑎 ∈ (ℤ𝑀))
177adantlr 477 . . . . 5 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))
18 fvi 5594 . . . . . 6 ((𝑥 + 𝑦) ∈ 𝑆 → ( I ‘(𝑥 + 𝑦)) = (𝑥 + 𝑦))
1914, 18syl 14 . . . . 5 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → ( I ‘(𝑥 + 𝑦)) = (𝑥 + 𝑦))
20 fvi 5594 . . . . . . 7 (𝑥𝑆 → ( I ‘𝑥) = 𝑥)
2120ad2antrl 490 . . . . . 6 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → ( I ‘𝑥) = 𝑥)
22 fvi 5594 . . . . . . 7 (𝑦𝑆 → ( I ‘𝑦) = 𝑦)
2322ad2antll 491 . . . . . 6 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → ( I ‘𝑦) = 𝑦)
2421, 23oveq12d 5915 . . . . 5 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (( I ‘𝑥)𝑄( I ‘𝑦)) = (𝑥𝑄𝑦))
2517, 19, 243eqtr4d 2232 . . . 4 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → ( I ‘(𝑥 + 𝑦)) = (( I ‘𝑥)𝑄( I ‘𝑦)))
26 fvi 5594 . . . . 5 ((𝐹𝑥) ∈ 𝑆 → ( I ‘(𝐹𝑥)) = (𝐹𝑥))
2715, 26syl 14 . . . 4 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → ( I ‘(𝐹𝑥)) = (𝐹𝑥))
288adantlr 477 . . . 4 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
2914, 15, 16, 25, 27, 15, 28seq3homo 10543 . . 3 ((𝜑𝑎 ∈ (ℤ𝑀)) → ( I ‘(seq𝑀( + , 𝐹)‘𝑎)) = (seq𝑀(𝑄, 𝐹)‘𝑎))
3013, 29eqtr3d 2224 . 2 ((𝜑𝑎 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹)‘𝑎) = (seq𝑀(𝑄, 𝐹)‘𝑎))
316, 10, 30eqfnfvd 5637 1 (𝜑 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160   I cid 4306  cfv 5235  (class class class)co 5897  cz 9284  cuz 9559  seqcseq 10478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-0id 7950  ax-rnegex 7951  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-ltadd 7958
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-frec 6417  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-inn 8951  df-n0 9208  df-z 9285  df-uz 9560  df-seqfrec 10479
This theorem is referenced by:  mulgpropdg  13121
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