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Theorem iseqss 9200
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 (𝜑𝑀 ∈ ℤ)
iseqss.t (𝜑𝑇𝑉)
iseqss.ss (𝜑𝑆𝑇)
iseqss.f ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
iseqss.pl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqss.plt ((𝜑 ∧ (𝑥𝑇𝑦𝑇)) → (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.
StepHypRef Expression
1 iseqss.m . . 3 (𝜑𝑀 ∈ ℤ)
2 iseqss.t . . . 4 (𝜑𝑇𝑉)
3 iseqss.ss . . . 4 (𝜑𝑆𝑇)
42, 3ssexd 3897 . . 3 (𝜑𝑆 ∈ V)
5 iseqss.f . . 3 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
6 iseqss.pl . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
71, 4, 5, 6iseqfn 9195 . 2 (𝜑 → seq𝑀( + , 𝐹, 𝑆) Fn (ℤ𝑀))
83sseld 2944 . . . . 5 (𝜑 → ((𝐹𝑥) ∈ 𝑆 → (𝐹𝑥) ∈ 𝑇))
98adantr 261 . . . 4 ((𝜑𝑥 ∈ (ℤ𝑀)) → ((𝐹𝑥) ∈ 𝑆 → (𝐹𝑥) ∈ 𝑇))
105, 9mpd 13 . . 3 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑇)
11 iseqss.plt . . 3 ((𝜑 ∧ (𝑥𝑇𝑦𝑇)) → (𝑥 + 𝑦) ∈ 𝑇)
121, 2, 10, 11iseqfn 9195 . 2 (𝜑 → seq𝑀( + , 𝐹, 𝑇) Fn (ℤ𝑀))
13 fveq2 5178 . . . . . 6 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
14 fveq2 5178 . . . . . 6 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑇)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑀))
1513, 14eqeq12d 2054 . . . . 5 (𝑤 = 𝑀 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq𝑀( + , 𝐹, 𝑇)‘𝑀)))
1615imbi2d 219 . . . 4 (𝑤 = 𝑀 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq𝑀( + , 𝐹, 𝑇)‘𝑀))))
17 fveq2 5178 . . . . . 6 (𝑤 = 𝑘 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑘))
18 fveq2 5178 . . . . . 6 (𝑤 = 𝑘 → (seq𝑀( + , 𝐹, 𝑇)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘))
1917, 18eqeq12d 2054 . . . . 5 (𝑤 = 𝑘 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘)))
2019imbi2d 219 . . . 4 (𝑤 = 𝑘 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘))))
21 fveq2 5178 . . . . . 6 (𝑤 = (𝑘 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)))
22 fveq2 5178 . . . . . 6 (𝑤 = (𝑘 + 1) → (seq𝑀( + , 𝐹, 𝑇)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)))
2321, 22eqeq12d 2054 . . . . 5 (𝑤 = (𝑘 + 1) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤) ↔ (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1))))
2423imbi2d 219 . . . 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 5178 . . . . . 6 (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))
26 fveq2 5178 . . . . . 6 (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑇)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑛))
2725, 26eqeq12d 2054 . . . . 5 (𝑤 = 𝑛 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq𝑀( + , 𝐹, 𝑇)‘𝑛)))
2827imbi2d 219 . . . 4 (𝑤 = 𝑛 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq𝑀( + , 𝐹, 𝑇)‘𝑛))))
291, 4, 5, 6iseq1 9196 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))
301, 2, 10, 11iseq1 9196 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑇)‘𝑀) = (𝐹𝑀))
3129, 30eqtr4d 2075 . . . . 5 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq𝑀( + , 𝐹, 𝑇)‘𝑀))
3231a1i 9 . . . 4 (𝑀 ∈ ℤ → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq𝑀( + , 𝐹, 𝑇)‘𝑀)))
33 oveq1 5519 . . . . . . 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 103 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝑀)) → 𝑘 ∈ (ℤ𝑀))
354adantr 261 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝑀)) → 𝑆 ∈ V)
365adantlr 446 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
376adantlr 446 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3834, 35, 36, 37iseqp1 9199 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))))
392adantr 261 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝑀)) → 𝑇𝑉)
4010adantlr 446 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑇)
4111adantlr 446 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (𝑥𝑇𝑦𝑇)) → (𝑥 + 𝑦) ∈ 𝑇)
4234, 39, 40, 41iseqp1 9199 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹, 𝑇)‘𝑘) + (𝐹‘(𝑘 + 1))))
4338, 42eqeq12d 2054 . . . . . . 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)))))
4433, 43syl5ibr 145 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1))))
4544expcom 109 . . . . 5 (𝑘 ∈ (ℤ𝑀) → (𝜑 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)))))
4645a2d 23 . . . 4 (𝑘 ∈ (ℤ𝑀) → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘)) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)))))
4716, 20, 24, 28, 32, 46uzind4 8529 . . 3 (𝑛 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq𝑀( + , 𝐹, 𝑇)‘𝑛)))
4847impcom 116 . 2 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq𝑀( + , 𝐹, 𝑇)‘𝑛))
497, 12, 48eqfnfvd 5268 1 (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wcel 1393  Vcvv 2557  wss 2917  cfv 4902  (class class class)co 5512  1c1 6888   + caddc 6890  cz 8243  cuz 8471  seqcseq 9185
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6973  ax-resscn 6974  ax-1cn 6975  ax-1re 6976  ax-icn 6977  ax-addcl 6978  ax-addrcl 6979  ax-mulcl 6980  ax-addcom 6982  ax-addass 6984  ax-distr 6986  ax-i2m1 6987  ax-0id 6990  ax-rnegex 6991  ax-cnre 6993  ax-pre-ltirr 6994  ax-pre-ltwlin 6995  ax-pre-lttrn 6996  ax-pre-ltadd 6998
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-i1p 6563  df-iplp 6564  df-iltp 6566  df-enr 6809  df-nr 6810  df-ltr 6813  df-0r 6814  df-1r 6815  df-0 6894  df-1 6895  df-r 6897  df-lt 6900  df-pnf 7060  df-mnf 7061  df-xr 7062  df-ltxr 7063  df-le 7064  df-sub 7182  df-neg 7183  df-inn 7913  df-n0 8180  df-z 8244  df-uz 8472  df-iseq 9186
This theorem is referenced by:  serige0  9226  serile  9227  iserile  9836  climserile  9839
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