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Theorem itcovalpclem2 47357
Description: Lemma 2 for itcovalpc 47358: 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 12478 . . . . . 6 β„•0 ∈ V
32mptex 7225 . . . . 5 (𝑛 ∈ β„•0 ↦ (𝑛 + 𝐢)) ∈ V
41, 3eqeltri 2830 . . . 4 𝐹 ∈ V
5 simpl 484 . . . 4 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 𝑦 ∈ β„•0)
6 simpr 486 . . . 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 47354 . . . 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 1469 . . 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 486 . . . . . . 7 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ 𝑛 ∈ β„•0)
10 simplr 768 . . . . . . . 8 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ 𝐢 ∈ β„•0)
115adantr 482 . . . . . . . 8 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ 𝑦 ∈ β„•0)
1210, 11nn0mulcld 12537 . . . . . . 7 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝐢 Β· 𝑦) ∈ β„•0)
139, 12nn0addcld 12536 . . . . . 6 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝑛 + (𝐢 Β· 𝑦)) ∈ β„•0)
14 eqidd 2734 . . . . . 6 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦))) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦))))
15 oveq1 7416 . . . . . . . . 9 (𝑛 = π‘š β†’ (𝑛 + 𝐢) = (π‘š + 𝐢))
1615cbvmptv 5262 . . . . . . . 8 (𝑛 ∈ β„•0 ↦ (𝑛 + 𝐢)) = (π‘š ∈ β„•0 ↦ (π‘š + 𝐢))
171, 16eqtri 2761 . . . . . . 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 7416 . . . . . 6 (π‘š = (𝑛 + (𝐢 Β· 𝑦)) β†’ (π‘š + 𝐢) = ((𝑛 + (𝐢 Β· 𝑦)) + 𝐢))
2013, 14, 18, 19fmptco 7127 . . . . 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 12534 . . . . . . . 8 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ 𝑛 ∈ β„‚)
2212nn0cnd 12534 . . . . . . . 8 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝐢 Β· 𝑦) ∈ β„‚)
2310nn0cnd 12534 . . . . . . . 8 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ 𝐢 ∈ β„‚)
2421, 22, 23addassd 11236 . . . . . . 7 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ ((𝑛 + (𝐢 Β· 𝑦)) + 𝐢) = (𝑛 + ((𝐢 Β· 𝑦) + 𝐢)))
25 nn0cn 12482 . . . . . . . . . . . . . 14 (𝐢 ∈ β„•0 β†’ 𝐢 ∈ β„‚)
2625mulridd 11231 . . . . . . . . . . . . 13 (𝐢 ∈ β„•0 β†’ (𝐢 Β· 1) = 𝐢)
2726adantl 483 . . . . . . . . . . . 12 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (𝐢 Β· 1) = 𝐢)
2827eqcomd 2739 . . . . . . . . . . 11 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 𝐢 = (𝐢 Β· 1))
2928oveq2d 7425 . . . . . . . . . 10 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ ((𝐢 Β· 𝑦) + 𝐢) = ((𝐢 Β· 𝑦) + (𝐢 Β· 1)))
30 simpr 486 . . . . . . . . . . . 12 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 𝐢 ∈ β„•0)
3130nn0cnd 12534 . . . . . . . . . . 11 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 𝐢 ∈ β„‚)
325nn0cnd 12534 . . . . . . . . . . 11 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 𝑦 ∈ β„‚)
33 1cnd 11209 . . . . . . . . . . 11 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 1 ∈ β„‚)
3431, 32, 33adddid 11238 . . . . . . . . . 10 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (𝐢 Β· (𝑦 + 1)) = ((𝐢 Β· 𝑦) + (𝐢 Β· 1)))
3529, 34eqtr4d 2776 . . . . . . . . 9 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ ((𝐢 Β· 𝑦) + 𝐢) = (𝐢 Β· (𝑦 + 1)))
3635oveq2d 7425 . . . . . . . 8 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (𝑛 + ((𝐢 Β· 𝑦) + 𝐢)) = (𝑛 + (𝐢 Β· (𝑦 + 1))))
3736adantr 482 . . . . . . 7 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝑛 + ((𝐢 Β· 𝑦) + 𝐢)) = (𝑛 + (𝐢 Β· (𝑦 + 1))))
3824, 37eqtrd 2773 . . . . . 6 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ ((𝑛 + (𝐢 Β· 𝑦)) + 𝐢) = (𝑛 + (𝐢 Β· (𝑦 + 1))))
3938mpteq2dva 5249 . . . . 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 2773 . . . 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 482 . . 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 2773 . 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 414 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 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ↦ cmpt 5232   ∘ ccom 5681  β€˜cfv 6544  (class class class)co 7409  1c1 11111   + caddc 11113   Β· cmul 11115  β„•0cn0 12472  IterCompcitco 47343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-seq 13967  df-itco 47345
This theorem is referenced by:  itcovalpc  47358
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