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Theorem itcovalpclem2 46910
Description: Lemma 2 for itcovalpc 46911: induction step. (Contributed by AV, 4-May-2024.)
Hypothesis
Ref Expression
itcovalpc.f 𝐹 = (𝑛 ∈ β„•0 ↦ (𝑛 + 𝐢))
Assertion
Ref Expression
itcovalpclem2 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦))) β†’ ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· (𝑦 + 1))))))
Distinct variable groups:   𝐢,𝑛   𝑦,𝑛
Allowed substitution hints:   𝐢(𝑦)   𝐹(𝑦,𝑛)
Proof of Theorem itcovalpclem2
Dummy variable π‘š is distinct from all other variables.
StepHypRef Expression
1 itcovalpc.f . . . . 5 𝐹 = (𝑛 ∈ β„•0 ↦ (𝑛 + 𝐢))
2 nn0ex 12443 . . . . . 6 β„•0 ∈ V
32mptex 7193 . . . . 5 (𝑛 ∈ β„•0 ↦ (𝑛 + 𝐢)) ∈ V
41, 3eqeltri 2828 . . . 4 𝐹 ∈ V
5 simpl 483 . . . 4 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 𝑦 ∈ β„•0)
6 simpr 485 . . . 4 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦)))) β†’ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦))))
7 itcovalsucov 46907 . . . 4 ((𝐹 ∈ V ∧ 𝑦 ∈ β„•0 ∧ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦)))) β†’ ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)) = (𝐹 ∘ (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦)))))
84, 5, 6, 7mp3an2ani 1468 . . 3 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦)))) β†’ ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)) = (𝐹 ∘ (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦)))))
9 simpr 485 . . . . . . 7 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ 𝑛 ∈ β„•0)
10 simplr 767 . . . . . . . 8 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ 𝐢 ∈ β„•0)
115adantr 481 . . . . . . . 8 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ 𝑦 ∈ β„•0)
1210, 11nn0mulcld 12502 . . . . . . 7 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝐢 Β· 𝑦) ∈ β„•0)
139, 12nn0addcld 12501 . . . . . 6 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝑛 + (𝐢 Β· 𝑦)) ∈ β„•0)
14 eqidd 2732 . . . . . 6 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦))) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦))))
15 oveq1 7384 . . . . . . . . 9 (𝑛 = π‘š β†’ (𝑛 + 𝐢) = (π‘š + 𝐢))
1615cbvmptv 5238 . . . . . . . 8 (𝑛 ∈ β„•0 ↦ (𝑛 + 𝐢)) = (π‘š ∈ β„•0 ↦ (π‘š + 𝐢))
171, 16eqtri 2759 . . . . . . 7 𝐹 = (π‘š ∈ β„•0 ↦ (π‘š + 𝐢))
1817a1i 11 . . . . . 6 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 𝐹 = (π‘š ∈ β„•0 ↦ (π‘š + 𝐢)))
19 oveq1 7384 . . . . . 6 (π‘š = (𝑛 + (𝐢 Β· 𝑦)) β†’ (π‘š + 𝐢) = ((𝑛 + (𝐢 Β· 𝑦)) + 𝐢))
2013, 14, 18, 19fmptco 7095 . . . . 5 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (𝐹 ∘ (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦)))) = (𝑛 ∈ β„•0 ↦ ((𝑛 + (𝐢 Β· 𝑦)) + 𝐢)))
219nn0cnd 12499 . . . . . . . 8 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ 𝑛 ∈ β„‚)
2212nn0cnd 12499 . . . . . . . 8 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝐢 Β· 𝑦) ∈ β„‚)
2310nn0cnd 12499 . . . . . . . 8 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ 𝐢 ∈ β„‚)
2421, 22, 23addassd 11201 . . . . . . 7 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ ((𝑛 + (𝐢 Β· 𝑦)) + 𝐢) = (𝑛 + ((𝐢 Β· 𝑦) + 𝐢)))
25 nn0cn 12447 . . . . . . . . . . . . . 14 (𝐢 ∈ β„•0 β†’ 𝐢 ∈ β„‚)
2625mulridd 11196 . . . . . . . . . . . . 13 (𝐢 ∈ β„•0 β†’ (𝐢 Β· 1) = 𝐢)
2726adantl 482 . . . . . . . . . . . 12 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (𝐢 Β· 1) = 𝐢)
2827eqcomd 2737 . . . . . . . . . . 11 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 𝐢 = (𝐢 Β· 1))
2928oveq2d 7393 . . . . . . . . . 10 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ ((𝐢 Β· 𝑦) + 𝐢) = ((𝐢 Β· 𝑦) + (𝐢 Β· 1)))
30 simpr 485 . . . . . . . . . . . 12 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 𝐢 ∈ β„•0)
3130nn0cnd 12499 . . . . . . . . . . 11 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 𝐢 ∈ β„‚)
325nn0cnd 12499 . . . . . . . . . . 11 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 𝑦 ∈ β„‚)
33 1cnd 11174 . . . . . . . . . . 11 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 1 ∈ β„‚)
3431, 32, 33adddid 11203 . . . . . . . . . 10 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (𝐢 Β· (𝑦 + 1)) = ((𝐢 Β· 𝑦) + (𝐢 Β· 1)))
3529, 34eqtr4d 2774 . . . . . . . . 9 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ ((𝐢 Β· 𝑦) + 𝐢) = (𝐢 Β· (𝑦 + 1)))
3635oveq2d 7393 . . . . . . . 8 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (𝑛 + ((𝐢 Β· 𝑦) + 𝐢)) = (𝑛 + (𝐢 Β· (𝑦 + 1))))
3736adantr 481 . . . . . . 7 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝑛 + ((𝐢 Β· 𝑦) + 𝐢)) = (𝑛 + (𝐢 Β· (𝑦 + 1))))
3824, 37eqtrd 2771 . . . . . 6 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ ((𝑛 + (𝐢 Β· 𝑦)) + 𝐢) = (𝑛 + (𝐢 Β· (𝑦 + 1))))
3938mpteq2dva 5225 . . . . 5 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (𝑛 ∈ β„•0 ↦ ((𝑛 + (𝐢 Β· 𝑦)) + 𝐢)) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· (𝑦 + 1)))))
4020, 39eqtrd 2771 . . . 4 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (𝐹 ∘ (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦)))) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· (𝑦 + 1)))))
4140adantr 481 . . 3 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦)))) β†’ (𝐹 ∘ (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦)))) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· (𝑦 + 1)))))
428, 41eqtrd 2771 . 2 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦)))) β†’ ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· (𝑦 + 1)))))
4342ex 413 1 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦))) β†’ ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· (𝑦 + 1))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3459   ↦ cmpt 5208   ∘ ccom 5657  β€˜cfv 6516  (class class class)co 7377  1c1 11076   + caddc 11078   Β· cmul 11080  β„•0cn0 12437  IterCompcitco 46896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-2nd 7942  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-er 8670  df-en 8906  df-dom 8907  df-sdom 8908  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11411  df-neg 11412  df-nn 12178  df-n0 12438  df-z 12524  df-uz 12788  df-seq 13932  df-itco 46898
This theorem is referenced by:  itcovalpc  46911
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