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Theorem iseqsst 9761
Description: Specifying a larger universe for seq. As long as 𝐹 and + are closed over 𝑆, then any class which contains 𝑆 can be used as the last argument to seq. (Contributed by Jim Kingdon, 28-Apr-2022.)
Hypotheses
Ref Expression
iseqsst.m (𝜑𝑀 ∈ ℤ)
iseqsst.ss (𝜑𝑆𝑇)
iseqsst.f ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
iseqsst.pl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
Assertion
Ref Expression
iseqsst (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

Proof of Theorem iseqsst
Dummy variables 𝑘 𝑤 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2083 . . . 4 (ℤ𝑀) = (ℤ𝑀)
2 iseqsst.m . . . 4 (𝜑𝑀 ∈ ℤ)
3 iseqsst.f . . . 4 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
4 iseqsst.pl . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
51, 2, 3, 4iseqfcl 9754 . . 3 (𝜑 → seq𝑀( + , 𝐹, 𝑆):(ℤ𝑀)⟶𝑆)
6 ffn 5114 . . 3 (seq𝑀( + , 𝐹, 𝑆):(ℤ𝑀)⟶𝑆 → seq𝑀( + , 𝐹, 𝑆) Fn (ℤ𝑀))
75, 6syl 14 . 2 (𝜑 → seq𝑀( + , 𝐹, 𝑆) Fn (ℤ𝑀))
8 iseqsst.ss . . . 4 (𝜑𝑆𝑇)
91, 2, 3, 4, 8iseqfclt 9755 . . 3 (𝜑 → seq𝑀( + , 𝐹, 𝑇):(ℤ𝑀)⟶𝑆)
10 ffn 5114 . . 3 (seq𝑀( + , 𝐹, 𝑇):(ℤ𝑀)⟶𝑆 → seq𝑀( + , 𝐹, 𝑇) Fn (ℤ𝑀))
119, 10syl 14 . 2 (𝜑 → seq𝑀( + , 𝐹, 𝑇) Fn (ℤ𝑀))
12 fveq2 5253 . . . . . 6 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
13 fveq2 5253 . . . . . 6 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑇)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑀))
1412, 13eqeq12d 2097 . . . . 5 (𝑤 = 𝑀 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq𝑀( + , 𝐹, 𝑇)‘𝑀)))
1514imbi2d 228 . . . 4 (𝑤 = 𝑀 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq𝑀( + , 𝐹, 𝑇)‘𝑀))))
16 fveq2 5253 . . . . . 6 (𝑤 = 𝑘 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑘))
17 fveq2 5253 . . . . . 6 (𝑤 = 𝑘 → (seq𝑀( + , 𝐹, 𝑇)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘))
1816, 17eqeq12d 2097 . . . . 5 (𝑤 = 𝑘 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘)))
1918imbi2d 228 . . . 4 (𝑤 = 𝑘 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘))))
20 fveq2 5253 . . . . . 6 (𝑤 = (𝑘 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)))
21 fveq2 5253 . . . . . 6 (𝑤 = (𝑘 + 1) → (seq𝑀( + , 𝐹, 𝑇)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)))
2220, 21eqeq12d 2097 . . . . 5 (𝑤 = (𝑘 + 1) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤) ↔ (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1))))
2322imbi2d 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)))))
24 fveq2 5253 . . . . . 6 (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))
25 fveq2 5253 . . . . . 6 (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑇)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑛))
2624, 25eqeq12d 2097 . . . . 5 (𝑤 = 𝑛 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq𝑀( + , 𝐹, 𝑇)‘𝑛)))
2726imbi2d 228 . . . 4 (𝑤 = 𝑛 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq𝑀( + , 𝐹, 𝑇)‘𝑛))))
282, 3, 4iseq1 9752 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))
292, 3, 4, 8iseq1t 9753 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑇)‘𝑀) = (𝐹𝑀))
3028, 29eqtr4d 2118 . . . . 5 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq𝑀( + , 𝐹, 𝑇)‘𝑀))
3130a1i 9 . . . 4 (𝑀 ∈ ℤ → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq𝑀( + , 𝐹, 𝑇)‘𝑀)))
32 oveq1 5598 . . . . . . 7 ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))) = ((seq𝑀( + , 𝐹, 𝑇)‘𝑘) + (𝐹‘(𝑘 + 1))))
33 simpr 108 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝑀)) → 𝑘 ∈ (ℤ𝑀))
343adantlr 461 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
354adantlr 461 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3633, 34, 35iseqp1 9757 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))))
378adantr 270 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝑀)) → 𝑆𝑇)
3833, 34, 35, 37iseqp1t 9758 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹, 𝑇)‘𝑘) + (𝐹‘(𝑘 + 1))))
3936, 38eqeq12d 2097 . . . . . . 7 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)) ↔ ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))) = ((seq𝑀( + , 𝐹, 𝑇)‘𝑘) + (𝐹‘(𝑘 + 1)))))
4032, 39syl5ibr 154 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1))))
4140expcom 114 . . . . 5 (𝑘 ∈ (ℤ𝑀) → (𝜑 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)))))
4241a2d 26 . . . 4 (𝑘 ∈ (ℤ𝑀) → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘)) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)))))
4315, 19, 23, 27, 31, 42uzind4 8971 . . 3 (𝑛 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq𝑀( + , 𝐹, 𝑇)‘𝑛)))
4443impcom 123 . 2 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq𝑀( + , 𝐹, 𝑇)‘𝑛))
457, 11, 44eqfnfvd 5345 1 (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  wss 2984   Fn wfn 4964  wf 4965  cfv 4969  (class class class)co 5591  1c1 7254   + caddc 7256  cz 8646  cuz 8914  seqcseq 9740
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366  ax-cnex 7339  ax-resscn 7340  ax-1cn 7341  ax-1re 7342  ax-icn 7343  ax-addcl 7344  ax-addrcl 7345  ax-mulcl 7346  ax-addcom 7348  ax-addass 7350  ax-distr 7352  ax-i2m1 7353  ax-0lt1 7354  ax-0id 7356  ax-rnegex 7357  ax-cnre 7359  ax-pre-ltirr 7360  ax-pre-ltwlin 7361  ax-pre-lttrn 7362  ax-pre-ltadd 7364
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-ilim 4160  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-riota 5547  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-frec 6088  df-pnf 7427  df-mnf 7428  df-xr 7429  df-ltxr 7430  df-le 7431  df-sub 7558  df-neg 7559  df-inn 8317  df-n0 8566  df-z 8647  df-uz 8915  df-iseq 9741
This theorem is referenced by: (None)
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