Theorem seqfeq3 10915

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	⊢ (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))

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.

Step	Hyp	Ref	Expression
1		eqid 2234	. . . 4 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
2		seqfeq3.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		seqfeq3.f	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
4		seqfeq3.cl	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
5	1, 2, 3, 4	seqf 10850	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶𝑆)
6	5	ffnd 5514	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀))
7		seqfeq3.id	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))
8	7, 4	eqeltrrd 2312	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
9	1, 2, 3, 8	seqf 10850	. . 3 ⊢ (𝜑 → seq𝑀(𝑄, 𝐹):(ℤ≥‘𝑀)⟶𝑆)
10	9	ffnd 5514	. 2 ⊢ (𝜑 → seq𝑀(𝑄, 𝐹) Fn (ℤ≥‘𝑀))
11	5	ffvelcdmda 5817	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹)‘𝑎) ∈ 𝑆)
12		fvi 5739	. . . 4 ⊢ ((seq𝑀( + , 𝐹)‘𝑎) ∈ 𝑆 → ( I ‘(seq𝑀( + , 𝐹)‘𝑎)) = (seq𝑀( + , 𝐹)‘𝑎))
13	11, 12	syl 14	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → ( I ‘(seq𝑀( + , 𝐹)‘𝑎)) = (seq𝑀( + , 𝐹)‘𝑎))
14	4	adantlr 477	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
15	3	adantlr 477	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
16		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → 𝑎 ∈ (ℤ≥‘𝑀))
17	7	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))
18		fvi 5739	. . . . . 6 ⊢ ((𝑥 + 𝑦) ∈ 𝑆 → ( I ‘(𝑥 + 𝑦)) = (𝑥 + 𝑦))
19	14, 18	syl 14	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ( I ‘(𝑥 + 𝑦)) = (𝑥 + 𝑦))
20		fvi 5739	. . . . . . 7 ⊢ (𝑥 ∈ 𝑆 → ( I ‘𝑥) = 𝑥)
21	20	ad2antrl 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ( I ‘𝑥) = 𝑥)
22		fvi 5739	. . . . . . 7 ⊢ (𝑦 ∈ 𝑆 → ( I ‘𝑦) = 𝑦)
23	22	ad2antll 491	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ( I ‘𝑦) = 𝑦)
24	21, 23	oveq12d 6076	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (( I ‘𝑥)𝑄( I ‘𝑦)) = (𝑥𝑄𝑦))
25	17, 19, 24	3eqtr4d 2277	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ( I ‘(𝑥 + 𝑦)) = (( I ‘𝑥)𝑄( I ‘𝑦)))
26		fvi 5739	. . . . 5 ⊢ ((𝐹‘𝑥) ∈ 𝑆 → ( I ‘(𝐹‘𝑥)) = (𝐹‘𝑥))
27	15, 26	syl 14	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ( I ‘(𝐹‘𝑥)) = (𝐹‘𝑥))
28	8	adantlr 477	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
29	14, 15, 16, 25, 27, 15, 28	seq3homo 10913	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → ( I ‘(seq𝑀( + , 𝐹)‘𝑎)) = (seq𝑀(𝑄, 𝐹)‘𝑎))
30	13, 29	eqtr3d 2269	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹)‘𝑎) = (seq𝑀(𝑄, 𝐹)‘𝑎))
31	6, 10, 30	eqfnfvd 5783	1 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205   I cid 4414  ‘cfv 5357  (class class class)co 6058  ℤcz 9594  ℤ_≥cuz 9871  seqcseq 10833

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-seqfrec 10834

This theorem is referenced by:  mulgpropdg  13917