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Theorem itcovalpclem2 46747
Description: Lemma 2 for itcovalpc 46748: induction step. (Contributed by AV, 4-May-2024.)
Hypothesis
Ref Expression
itcovalpc.f 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))
Assertion
Ref Expression
itcovalpclem2 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))
Distinct variable groups:   𝐶,𝑛   𝑦,𝑛
Allowed substitution hints:   𝐶(𝑦)   𝐹(𝑦,𝑛)

Proof of Theorem itcovalpclem2
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 itcovalpc.f . . . . 5 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))
2 nn0ex 12419 . . . . . 6 0 ∈ V
32mptex 7173 . . . . 5 (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶)) ∈ V
41, 3eqeltri 2834 . . . 4 𝐹 ∈ V
5 simpl 483 . . . 4 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → 𝑦 ∈ ℕ0)
6 simpr 485 . . . 4 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))))
7 itcovalsucov 46744 . . . 4 ((𝐹 ∈ V ∧ 𝑦 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))))
84, 5, 6, 7mp3an2ani 1468 . . 3 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))))
9 simpr 485 . . . . . . 7 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
10 simplr 767 . . . . . . . 8 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝐶 ∈ ℕ0)
115adantr 481 . . . . . . . 8 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝑦 ∈ ℕ0)
1210, 11nn0mulcld 12478 . . . . . . 7 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝐶 · 𝑦) ∈ ℕ0)
139, 12nn0addcld 12477 . . . . . 6 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑛 + (𝐶 · 𝑦)) ∈ ℕ0)
14 eqidd 2737 . . . . . 6 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))))
15 oveq1 7364 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑛 + 𝐶) = (𝑚 + 𝐶))
1615cbvmptv 5218 . . . . . . . 8 (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶)) = (𝑚 ∈ ℕ0 ↦ (𝑚 + 𝐶))
171, 16eqtri 2764 . . . . . . 7 𝐹 = (𝑚 ∈ ℕ0 ↦ (𝑚 + 𝐶))
1817a1i 11 . . . . . 6 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → 𝐹 = (𝑚 ∈ ℕ0 ↦ (𝑚 + 𝐶)))
19 oveq1 7364 . . . . . 6 (𝑚 = (𝑛 + (𝐶 · 𝑦)) → (𝑚 + 𝐶) = ((𝑛 + (𝐶 · 𝑦)) + 𝐶))
2013, 14, 18, 19fmptco 7075 . . . . 5 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) = (𝑛 ∈ ℕ0 ↦ ((𝑛 + (𝐶 · 𝑦)) + 𝐶)))
219nn0cnd 12475 . . . . . . . 8 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℂ)
2212nn0cnd 12475 . . . . . . . 8 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝐶 · 𝑦) ∈ ℂ)
2310nn0cnd 12475 . . . . . . . 8 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝐶 ∈ ℂ)
2421, 22, 23addassd 11177 . . . . . . 7 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑛 + (𝐶 · 𝑦)) + 𝐶) = (𝑛 + ((𝐶 · 𝑦) + 𝐶)))
25 nn0cn 12423 . . . . . . . . . . . . . 14 (𝐶 ∈ ℕ0𝐶 ∈ ℂ)
2625mulid1d 11172 . . . . . . . . . . . . 13 (𝐶 ∈ ℕ0 → (𝐶 · 1) = 𝐶)
2726adantl 482 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (𝐶 · 1) = 𝐶)
2827eqcomd 2742 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → 𝐶 = (𝐶 · 1))
2928oveq2d 7373 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → ((𝐶 · 𝑦) + 𝐶) = ((𝐶 · 𝑦) + (𝐶 · 1)))
30 simpr 485 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → 𝐶 ∈ ℕ0)
3130nn0cnd 12475 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → 𝐶 ∈ ℂ)
325nn0cnd 12475 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → 𝑦 ∈ ℂ)
33 1cnd 11150 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → 1 ∈ ℂ)
3431, 32, 33adddid 11179 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (𝐶 · (𝑦 + 1)) = ((𝐶 · 𝑦) + (𝐶 · 1)))
3529, 34eqtr4d 2779 . . . . . . . . 9 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → ((𝐶 · 𝑦) + 𝐶) = (𝐶 · (𝑦 + 1)))
3635oveq2d 7373 . . . . . . . 8 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (𝑛 + ((𝐶 · 𝑦) + 𝐶)) = (𝑛 + (𝐶 · (𝑦 + 1))))
3736adantr 481 . . . . . . 7 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑛 + ((𝐶 · 𝑦) + 𝐶)) = (𝑛 + (𝐶 · (𝑦 + 1))))
3824, 37eqtrd 2776 . . . . . 6 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑛 + (𝐶 · 𝑦)) + 𝐶) = (𝑛 + (𝐶 · (𝑦 + 1))))
3938mpteq2dva 5205 . . . . 5 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ ((𝑛 + (𝐶 · 𝑦)) + 𝐶)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
4020, 39eqtrd 2776 . . . 4 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
4140adantr 481 . . 3 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
428, 41eqtrd 2776 . 2 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
4342ex 413 1 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  cmpt 5188  ccom 5637  cfv 6496  (class class class)co 7357  1c1 11052   + caddc 11054   · cmul 11056  0cn0 12413  IterCompcitco 46733
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-seq 13907  df-itco 46735
This theorem is referenced by:  itcovalpc  46748
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