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Theorem iccpartimp 46075
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 46074 . . 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 6891 . . . . . . 7 (𝑖 = 𝐼 β†’ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜πΌ))
3 fvoveq1 7431 . . . . . . 7 (𝑖 = 𝐼 β†’ (π‘ƒβ€˜(𝑖 + 1)) = (π‘ƒβ€˜(𝐼 + 1)))
42, 3breq12d 5161 . . . . . 6 (𝑖 = 𝐼 β†’ ((π‘ƒβ€˜π‘–) < (π‘ƒβ€˜(𝑖 + 1)) ↔ (π‘ƒβ€˜πΌ) < (π‘ƒβ€˜(𝐼 + 1))))
54rspccv 3609 . . . . 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 482 . . . 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 483 . . . 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 526 . . 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 252 . 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 1111 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7408   ↑m cmap 8819  0cc0 11109  1c1 11110   + caddc 11112  β„*cxr 11246   < clt 11247  β„•cn 12211  ...cfz 13483  ..^cfzo 13626  RePartciccp 46071
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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-iccp 46072
This theorem is referenced by:  iccpartgtprec  46078  iccpartipre  46079  iccpartiltu  46080  iccpartigtl  46081  iccpartlt  46082  iccpartgt  46085
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