Theorem iseqss 9446

Description: Specifying a larger universe for seq. As long as 𝐹 and + are closed over 𝑆, then any set which contains 𝑆 can be used as the last argument to seq. This theorem does not allow 𝑇 to be a proper class, however. It also currently requires that + be closed over 𝑇 (as well as 𝑆). (Contributed by Jim Kingdon, 18-Aug-2021.)

Hypotheses

Ref	Expression
iseqss.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
iseqss.t	⊢ (𝜑 → 𝑇 ∈ 𝑉)
iseqss.ss	⊢ (𝜑 → 𝑆 ⊆ 𝑇)
iseqss.f	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
iseqss.pl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqss.plt	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑇)) → (𝑥 + 𝑦) ∈ 𝑇)

Assertion

Ref	Expression
iseqss	⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))

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

Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem iseqss

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

Step	Hyp	Ref	Expression
1		iseqss.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2		iseqss.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
3		iseqss.ss	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
4	2, 3	ssexd 3918	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
5		iseqss.f	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
6		iseqss.pl	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
7	1, 4, 5, 6	iseqfn 9441	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) Fn (ℤ≥‘𝑀))
8	3	sseld 2998	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑥) ∈ 𝑆 → (𝐹‘𝑥) ∈ 𝑇))
9	8	adantr 270	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝐹‘𝑥) ∈ 𝑆 → (𝐹‘𝑥) ∈ 𝑇))
10	5, 9	mpd 13	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑇)
11		iseqss.plt	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑇)) → (𝑥 + 𝑦) ∈ 𝑇)
12	1, 2, 10, 11	iseqfn 9441	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑇) Fn (ℤ≥‘𝑀))
13		fveq2 5198	. . . . . 6 ⊢ (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
14		fveq2 5198	. . . . . 6 ⊢ (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑇)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑀))
15	13, 14	eqeq12d 2095	. . . . 5 ⊢ (𝑤 = 𝑀 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq𝑀( + , 𝐹, 𝑇)‘𝑀)))
16	15	imbi2d 228	. . . 4 ⊢ (𝑤 = 𝑀 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq𝑀( + , 𝐹, 𝑇)‘𝑀))))
17		fveq2 5198	. . . . . 6 ⊢ (𝑤 = 𝑘 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑘))
18		fveq2 5198	. . . . . 6 ⊢ (𝑤 = 𝑘 → (seq𝑀( + , 𝐹, 𝑇)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘))
19	17, 18	eqeq12d 2095	. . . . 5 ⊢ (𝑤 = 𝑘 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘)))
20	19	imbi2d 228	. . . 4 ⊢ (𝑤 = 𝑘 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘))))
21		fveq2 5198	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)))
22		fveq2 5198	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → (seq𝑀( + , 𝐹, 𝑇)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)))
23	21, 22	eqeq12d 2095	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤) ↔ (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1))))
24	23	imbi2d 228	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)))))
25		fveq2 5198	. . . . . 6 ⊢ (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))
26		fveq2 5198	. . . . . 6 ⊢ (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑇)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑛))
27	25, 26	eqeq12d 2095	. . . . 5 ⊢ (𝑤 = 𝑛 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq𝑀( + , 𝐹, 𝑇)‘𝑛)))
28	27	imbi2d 228	. . . 4 ⊢ (𝑤 = 𝑛 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq𝑀( + , 𝐹, 𝑇)‘𝑛))))
29	1, 4, 5, 6	iseq1 9442	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹‘𝑀))
30	1, 2, 10, 11	iseq1 9442	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑇)‘𝑀) = (𝐹‘𝑀))
31	29, 30	eqtr4d 2116	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq𝑀( + , 𝐹, 𝑇)‘𝑀))
32	31	a1i 9	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq𝑀( + , 𝐹, 𝑇)‘𝑀)))
33		oveq1 5539	. . . . . . 7 ⊢ ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))) = ((seq𝑀( + , 𝐹, 𝑇)‘𝑘) + (𝐹‘(𝑘 + 1))))
34		simpr 108	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ (ℤ≥‘𝑀))
35	4	adantr 270	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑆 ∈ V)
36	5	adantlr 460	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
37	6	adantlr 460	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
38	34, 35, 36, 37	iseqp1 9445	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))))
39	2	adantr 270	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑇 ∈ 𝑉)
40	10	adantlr 460	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑇)
41	11	adantlr 460	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑇)) → (𝑥 + 𝑦) ∈ 𝑇)
42	34, 39, 40, 41	iseqp1 9445	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹, 𝑇)‘𝑘) + (𝐹‘(𝑘 + 1))))
43	38, 42	eqeq12d 2095	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)) ↔ ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))) = ((seq𝑀( + , 𝐹, 𝑇)‘𝑘) + (𝐹‘(𝑘 + 1)))))
44	33, 43	syl5ibr 154	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1))))
45	44	expcom 114	. . . . 5 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜑 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)))))
46	45	a2d 26	. . . 4 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘)) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)))))
47	16, 20, 24, 28, 32, 46	uzind4 8676	. . 3 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq𝑀( + , 𝐹, 𝑇)‘𝑛)))
48	47	impcom 123	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq𝑀( + , 𝐹, 𝑇)‘𝑛))
49	7, 12, 48	eqfnfvd 5289	1 ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ⊆ wss 2973  ‘cfv 4922  (class class class)co 5532  1c1 6982   + caddc 6984  ℤcz 8351  ℤ_≥cuz 8619  seqcseq 9431

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432

This theorem is referenced by:  serige0  9473  serile  9474  iserile  10180  climserile  10183