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Theorem iccpartimp 45699
Description: Implications for a class being a partition. (Contributed by AV, 11-Jul-2020.)
Assertion
Ref Expression
iccpartimp ((𝑀 ∈ β„• ∧ 𝑃 ∈ (RePartβ€˜π‘€) ∧ 𝐼 ∈ (0..^𝑀)) β†’ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (π‘ƒβ€˜πΌ) < (π‘ƒβ€˜(𝐼 + 1))))

Proof of Theorem iccpartimp
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 iccpart 45698 . . 3 (𝑀 ∈ β„• β†’ (𝑃 ∈ (RePartβ€˜π‘€) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘ƒβ€˜π‘–) < (π‘ƒβ€˜(𝑖 + 1)))))
2 fveq2 6846 . . . . . . 7 (𝑖 = 𝐼 β†’ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜πΌ))
3 fvoveq1 7384 . . . . . . 7 (𝑖 = 𝐼 β†’ (π‘ƒβ€˜(𝑖 + 1)) = (π‘ƒβ€˜(𝐼 + 1)))
42, 3breq12d 5122 . . . . . 6 (𝑖 = 𝐼 β†’ ((π‘ƒβ€˜π‘–) < (π‘ƒβ€˜(𝑖 + 1)) ↔ (π‘ƒβ€˜πΌ) < (π‘ƒβ€˜(𝐼 + 1))))
54rspccv 3580 . . . . 5 (βˆ€π‘– ∈ (0..^𝑀)(π‘ƒβ€˜π‘–) < (π‘ƒβ€˜(𝑖 + 1)) β†’ (𝐼 ∈ (0..^𝑀) β†’ (π‘ƒβ€˜πΌ) < (π‘ƒβ€˜(𝐼 + 1))))
65adantl 483 . . . 4 ((𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘ƒβ€˜π‘–) < (π‘ƒβ€˜(𝑖 + 1))) β†’ (𝐼 ∈ (0..^𝑀) β†’ (π‘ƒβ€˜πΌ) < (π‘ƒβ€˜(𝐼 + 1))))
7 simpl 484 . . . 4 ((𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘ƒβ€˜π‘–) < (π‘ƒβ€˜(𝑖 + 1))) β†’ 𝑃 ∈ (ℝ* ↑m (0...𝑀)))
86, 7jctild 527 . . 3 ((𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘ƒβ€˜π‘–) < (π‘ƒβ€˜(𝑖 + 1))) β†’ (𝐼 ∈ (0..^𝑀) β†’ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (π‘ƒβ€˜πΌ) < (π‘ƒβ€˜(𝐼 + 1)))))
91, 8syl6bi 253 . 2 (𝑀 ∈ β„• β†’ (𝑃 ∈ (RePartβ€˜π‘€) β†’ (𝐼 ∈ (0..^𝑀) β†’ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (π‘ƒβ€˜πΌ) < (π‘ƒβ€˜(𝐼 + 1))))))
1093imp 1112 1 ((𝑀 ∈ β„• ∧ 𝑃 ∈ (RePartβ€˜π‘€) ∧ 𝐼 ∈ (0..^𝑀)) β†’ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (π‘ƒβ€˜πΌ) < (π‘ƒβ€˜(𝐼 + 1))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   class class class wbr 5109  β€˜cfv 6500  (class class class)co 7361   ↑m cmap 8771  0cc0 11059  1c1 11060   + caddc 11062  β„*cxr 11196   < clt 11197  β„•cn 12161  ...cfz 13433  ..^cfzo 13576  RePartciccp 45695
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-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-iccp 45696
This theorem is referenced by:  iccpartgtprec  45702  iccpartipre  45703  iccpartiltu  45704  iccpartigtl  45705  iccpartlt  45706  iccpartgt  45709
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