Theorem seqfeq3 10746

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 2229	. . . 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 10681	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶𝑆)
6	5	ffnd 5473	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀))
7		seqfeq3.id	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))
8	7, 4	eqeltrrd 2307	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
9	1, 2, 3, 8	seqf 10681	. . 3 ⊢ (𝜑 → seq𝑀(𝑄, 𝐹):(ℤ≥‘𝑀)⟶𝑆)
10	9	ffnd 5473	. 2 ⊢ (𝜑 → seq𝑀(𝑄, 𝐹) Fn (ℤ≥‘𝑀))
11	5	ffvelcdmda 5769	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹)‘𝑎) ∈ 𝑆)
12		fvi 5690	. . . 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 5690	. . . . . 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 5690	. . . . . . 7 ⊢ (𝑥 ∈ 𝑆 → ( I ‘𝑥) = 𝑥)
21	20	ad2antrl 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ( I ‘𝑥) = 𝑥)
22		fvi 5690	. . . . . . 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 6018	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (( I ‘𝑥)𝑄( I ‘𝑦)) = (𝑥𝑄𝑦))
25	17, 19, 24	3eqtr4d 2272	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ( I ‘(𝑥 + 𝑦)) = (( I ‘𝑥)𝑄( I ‘𝑦)))
26		fvi 5690	. . . . 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 10744	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → ( I ‘(seq𝑀( + , 𝐹)‘𝑎)) = (seq𝑀(𝑄, 𝐹)‘𝑎))
30	13, 29	eqtr3d 2264	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹)‘𝑎) = (seq𝑀(𝑄, 𝐹)‘𝑎))
31	6, 10, 30	eqfnfvd 5734	1 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200   I cid 4378  ‘cfv 5317  (class class class)co 6000  ℤcz 9442  ℤ_≥cuz 9718  seqcseq 10664

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-seqfrec 10665

This theorem is referenced by:  mulgpropdg  13696