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Theorem iseqsst 9850
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.

Together with df-seq3 9819 it can be used to convert between the df-iseq 9818 syntax and the df-seq3 9819 syntax (in many cases iseqseq3 9866 is an even more convenient way to do this).

(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 2088 . . . 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 9841 . . 3 (𝜑 → seq𝑀( + , 𝐹, 𝑆):(ℤ𝑀)⟶𝑆)
6 ffn 5147 . . 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 9842 . . 3 (𝜑 → seq𝑀( + , 𝐹, 𝑇):(ℤ𝑀)⟶𝑆)
10 ffn 5147 . . 3 (seq𝑀( + , 𝐹, 𝑇):(ℤ𝑀)⟶𝑆 → seq𝑀( + , 𝐹, 𝑇) Fn (ℤ𝑀))
119, 10syl 14 . 2 (𝜑 → seq𝑀( + , 𝐹, 𝑇) Fn (ℤ𝑀))
12 fveq2 5289 . . . . . 6 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
13 fveq2 5289 . . . . . 6 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑇)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑀))
1412, 13eqeq12d 2102 . . . . 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 5289 . . . . . 6 (𝑤 = 𝑘 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑘))
17 fveq2 5289 . . . . . 6 (𝑤 = 𝑘 → (seq𝑀( + , 𝐹, 𝑇)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑘))
1816, 17eqeq12d 2102 . . . . 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 5289 . . . . . 6 (𝑤 = (𝑘 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)))
21 fveq2 5289 . . . . . 6 (𝑤 = (𝑘 + 1) → (seq𝑀( + , 𝐹, 𝑇)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)))
2220, 21eqeq12d 2102 . . . . 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 5289 . . . . . 6 (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))
25 fveq2 5289 . . . . . 6 (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑇)‘𝑤) = (seq𝑀( + , 𝐹, 𝑇)‘𝑛))
2624, 25eqeq12d 2102 . . . . 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 9838 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))
292, 3, 4, 8iseq1t 9839 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑇)‘𝑀) = (𝐹𝑀))
3028, 29eqtr4d 2123 . . . . 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 5641 . . . . . . 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 9845 . . . . . . . 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 9846 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹, 𝑇)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹, 𝑇)‘𝑘) + (𝐹‘(𝑘 + 1))))
3936, 38eqeq12d 2102 . . . . . . 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 9045 . . 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 5384 1 (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  wss 2997   Fn wfn 4997  wf 4998  cfv 5002  (class class class)co 5634  1c1 7330   + caddc 7332  cz 8720  cuz 8988  seqcseq4 9816
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721  df-uz 8989  df-iseq 9818
This theorem is referenced by:  seq3clss  9851  seq3feq  9861  seq3shft2  9863  iseqseq3  9866
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