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Theorem itcovalpclem2 47444
Description: Lemma 2 for itcovalpc 47445: 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 12482 . . . . . 6 β„•0 ∈ V
32mptex 7226 . . . . 5 (𝑛 ∈ β„•0 ↦ (𝑛 + 𝐢)) ∈ V
41, 3eqeltri 2827 . . . 4 𝐹 ∈ V
5 simpl 481 . . . 4 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 𝑦 ∈ β„•0)
6 simpr 483 . . . 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 47441 . . . 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 1466 . . 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 483 . . . . . . 7 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ 𝑛 ∈ β„•0)
10 simplr 765 . . . . . . . 8 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ 𝐢 ∈ β„•0)
115adantr 479 . . . . . . . 8 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ 𝑦 ∈ β„•0)
1210, 11nn0mulcld 12541 . . . . . . 7 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝐢 Β· 𝑦) ∈ β„•0)
139, 12nn0addcld 12540 . . . . . 6 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝑛 + (𝐢 Β· 𝑦)) ∈ β„•0)
14 eqidd 2731 . . . . . 6 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦))) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦))))
15 oveq1 7418 . . . . . . . . 9 (𝑛 = π‘š β†’ (𝑛 + 𝐢) = (π‘š + 𝐢))
1615cbvmptv 5260 . . . . . . . 8 (𝑛 ∈ β„•0 ↦ (𝑛 + 𝐢)) = (π‘š ∈ β„•0 ↦ (π‘š + 𝐢))
171, 16eqtri 2758 . . . . . . 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 7418 . . . . . 6 (π‘š = (𝑛 + (𝐢 Β· 𝑦)) β†’ (π‘š + 𝐢) = ((𝑛 + (𝐢 Β· 𝑦)) + 𝐢))
2013, 14, 18, 19fmptco 7128 . . . . 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 12538 . . . . . . . 8 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ 𝑛 ∈ β„‚)
2212nn0cnd 12538 . . . . . . . 8 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝐢 Β· 𝑦) ∈ β„‚)
2310nn0cnd 12538 . . . . . . . 8 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ 𝐢 ∈ β„‚)
2421, 22, 23addassd 11240 . . . . . . 7 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ ((𝑛 + (𝐢 Β· 𝑦)) + 𝐢) = (𝑛 + ((𝐢 Β· 𝑦) + 𝐢)))
25 nn0cn 12486 . . . . . . . . . . . . . 14 (𝐢 ∈ β„•0 β†’ 𝐢 ∈ β„‚)
2625mulridd 11235 . . . . . . . . . . . . 13 (𝐢 ∈ β„•0 β†’ (𝐢 Β· 1) = 𝐢)
2726adantl 480 . . . . . . . . . . . 12 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (𝐢 Β· 1) = 𝐢)
2827eqcomd 2736 . . . . . . . . . . 11 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 𝐢 = (𝐢 Β· 1))
2928oveq2d 7427 . . . . . . . . . 10 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ ((𝐢 Β· 𝑦) + 𝐢) = ((𝐢 Β· 𝑦) + (𝐢 Β· 1)))
30 simpr 483 . . . . . . . . . . . 12 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 𝐢 ∈ β„•0)
3130nn0cnd 12538 . . . . . . . . . . 11 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 𝐢 ∈ β„‚)
325nn0cnd 12538 . . . . . . . . . . 11 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 𝑦 ∈ β„‚)
33 1cnd 11213 . . . . . . . . . . 11 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ 1 ∈ β„‚)
3431, 32, 33adddid 11242 . . . . . . . . . 10 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (𝐢 Β· (𝑦 + 1)) = ((𝐢 Β· 𝑦) + (𝐢 Β· 1)))
3529, 34eqtr4d 2773 . . . . . . . . 9 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ ((𝐢 Β· 𝑦) + 𝐢) = (𝐢 Β· (𝑦 + 1)))
3635oveq2d 7427 . . . . . . . 8 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (𝑛 + ((𝐢 Β· 𝑦) + 𝐢)) = (𝑛 + (𝐢 Β· (𝑦 + 1))))
3736adantr 479 . . . . . . 7 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝑛 + ((𝐢 Β· 𝑦) + 𝐢)) = (𝑛 + (𝐢 Β· (𝑦 + 1))))
3824, 37eqtrd 2770 . . . . . 6 (((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ ((𝑛 + (𝐢 Β· 𝑦)) + 𝐢) = (𝑛 + (𝐢 Β· (𝑦 + 1))))
3938mpteq2dva 5247 . . . . 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 2770 . . . 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 479 . . 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 2770 . 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 411 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 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472   ↦ cmpt 5230   ∘ ccom 5679  β€˜cfv 6542  (class class class)co 7411  1c1 11113   + caddc 11115   Β· cmul 11117  β„•0cn0 12476  IterCompcitco 47430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-seq 13971  df-itco 47432
This theorem is referenced by:  itcovalpc  47445
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