Theorem seq3id3 10741

Description: A sequence that consists entirely of "zeroes" sums to "zero". More precisely, a constant sequence with value an element which is a + -idempotent sums (or "+'s") to that element. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Jim Kingdon, 8-Apr-2023.)

Hypotheses

Ref	Expression
iseqid3s.1	⊢ (𝜑 → (𝑍 + 𝑍) = 𝑍)
iseqid3s.2	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
iseqid3s.3	⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = 𝑍)
iseqid3s.z	⊢ (𝜑 → 𝑍 ∈ 𝑆)
iseqid3s.f	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
iseqid3s.cl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)

Assertion

Ref	Expression
seq3id3	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)

Distinct variable groups:   𝑥,𝑦, +   𝑥,𝐹,𝑦   𝑥,𝑀,𝑦   𝜑,𝑥,𝑦   𝑥,𝑍,𝑦   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦

Proof of Theorem seq3id3

Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqid3s.2	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		eluzfz2 10224	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3		fveqeq2 5635	. . . . 5 ⊢ (𝑤 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑀) = 𝑍))
4	3	imbi2d 230	. . . 4 ⊢ (𝑤 = 𝑀 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = 𝑍)))
5		fveqeq2 5635	. . . . 5 ⊢ (𝑤 = 𝑘 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍))
6	5	imbi2d 230	. . . 4 ⊢ (𝑤 = 𝑘 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍)))
7		fveqeq2 5635	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍))
8	7	imbi2d 230	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
9		fveqeq2 5635	. . . . 5 ⊢ (𝑤 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑁) = 𝑍))
10	9	imbi2d 230	. . . 4 ⊢ (𝑤 = 𝑁 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)))
11		eluzel2 9723	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
12	1, 11	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
13		iseqid3s.f	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
14		iseqid3s.cl	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
15	12, 13, 14	seq3-1 10679	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
16		iseqid3s.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = 𝑍)
17	16	ralrimiva 2603	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) = 𝑍)
18		eluzfz1 10223	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
19		fveqeq2 5635	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) = 𝑍 ↔ (𝐹‘𝑀) = 𝑍))
20	19	rspcv 2903	. . . . . . . 8 ⊢ (𝑀 ∈ (𝑀...𝑁) → (∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) = 𝑍 → (𝐹‘𝑀) = 𝑍))
21	1, 18, 20	3syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) = 𝑍 → (𝐹‘𝑀) = 𝑍))
22	17, 21	mpd 13	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) = 𝑍)
23	15, 22	eqtrd 2262	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = 𝑍)
24	23	a1i 9	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = 𝑍))
25		elfzouz 10343	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (ℤ≥‘𝑀))
26	25	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ (ℤ≥‘𝑀))
27	13	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
28	14	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
29	26, 27, 28	seq3p1 10682	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
30	29	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
31		simpr 110	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍)
32		fveqeq2 5635	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → ((𝐹‘𝑥) = 𝑍 ↔ (𝐹‘(𝑘 + 1)) = 𝑍))
33	17	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) = 𝑍)
34		fzofzp1 10428	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
35	34	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
36	32, 33, 35	rspcdva 2912	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑘 + 1)) = 𝑍)
37	36	adantr 276	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (𝐹‘(𝑘 + 1)) = 𝑍)
38	31, 37	oveq12d 6018	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))) = (𝑍 + 𝑍))
39		iseqid3s.1	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 + 𝑍) = 𝑍)
40	39	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (𝑍 + 𝑍) = 𝑍)
41	30, 38, 40	3eqtrd 2266	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)
42	41	ex 115	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍))
43	42	expcom 116	. . . . 5 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( + , 𝐹)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
44	43	a2d 26	. . . 4 ⊢ (𝑘 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (𝜑 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
45	4, 6, 8, 10, 24, 44	fzind2 10440	. . 3 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍))
46	1, 2, 45	3syl 17	. 2 ⊢ (𝜑 → (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍))
47	46	pm2.43i 49	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ‘cfv 5317  (class class class)co 6000  1c1 7996   + caddc 7998  ℤcz 9442  ℤ_≥cuz 9718  ...cfz 10200  ..^cfzo 10334  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-fz 10201  df-fzo 10335  df-seqfrec 10665

This theorem is referenced by:  seq3id  10742  ser0  10750  prodf1  12048  mulgnn0z  13681  lgsval2lem  15683