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Theorem sticksstones8 41061
Description: Establish mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 1-Oct-2024.)
Hypotheses
Ref Expression
sticksstones8.1 (πœ‘ β†’ 𝑁 ∈ β„•0)
sticksstones8.2 (πœ‘ β†’ 𝐾 ∈ β„•0)
sticksstones8.3 𝐹 = (π‘Ž ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))))
sticksstones8.4 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘”β€˜π‘–) = 𝑁)}
sticksstones8.5 𝐡 = {𝑓 ∣ (𝑓:(1...𝐾)⟢(1...(𝑁 + 𝐾)) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)))}
Assertion
Ref Expression
sticksstones8 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
Distinct variable groups:   𝐴,π‘Ž,𝑗,𝑙,π‘₯,𝑦   𝐡,π‘Ž   𝑗,𝐾,𝑙,𝑓,π‘₯,𝑦   𝑔,𝐾,𝑖   𝑓,𝑁,𝑗   𝑔,𝑁   𝑓,π‘Ž   𝑔,π‘Ž,𝑖   πœ‘,π‘Ž,𝑗,𝑙   𝑖,𝑙   πœ‘,π‘₯,𝑦
Allowed substitution hints:   πœ‘(𝑓,𝑔,𝑖)   𝐴(𝑓,𝑔,𝑖)   𝐡(π‘₯,𝑦,𝑓,𝑔,𝑖,𝑗,𝑙)   𝐹(π‘₯,𝑦,𝑓,𝑔,𝑖,𝑗,π‘Ž,𝑙)   𝐾(π‘Ž)   𝑁(π‘₯,𝑦,𝑖,π‘Ž,𝑙)

Proof of Theorem sticksstones8
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2733 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(π‘Žβ€˜π‘™))) = (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(π‘Žβ€˜π‘™))))
2 simpr 485 . . . . . . . . . 10 (((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) β†’ 𝑒 = 𝑗)
32oveq2d 7427 . . . . . . . . . . 11 (((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) β†’ (1...𝑒) = (1...𝑗))
43sumeq1d 15649 . . . . . . . . . 10 (((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) β†’ Σ𝑙 ∈ (1...𝑒)(π‘Žβ€˜π‘™) = Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))
52, 4oveq12d 7429 . . . . . . . . 9 (((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) β†’ (𝑒 + Σ𝑙 ∈ (1...𝑒)(π‘Žβ€˜π‘™)) = (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))
6 simp3 1138 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ 𝑗 ∈ (1...𝐾))
7 ovexd 7446 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)) ∈ V)
81, 5, 6, 7fvmptd 7005 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ ((𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(π‘Žβ€˜π‘™)))β€˜π‘—) = (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))
9 sticksstones8.1 . . . . . . . . . 10 (πœ‘ β†’ 𝑁 ∈ β„•0)
1093ad2ant1 1133 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ 𝑁 ∈ β„•0)
11 sticksstones8.2 . . . . . . . . . 10 (πœ‘ β†’ 𝐾 ∈ β„•0)
12113ad2ant1 1133 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ 𝐾 ∈ β„•0)
13 simpr 485 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ π‘Ž ∈ 𝐴)
14 sticksstones8.4 . . . . . . . . . . . . . . 15 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘”β€˜π‘–) = 𝑁)}
1514a1i 11 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘”β€˜π‘–) = 𝑁)})
1615eqcomd 2738 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘”β€˜π‘–) = 𝑁)} = 𝐴)
1713, 16eleqtrrd 2836 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ π‘Ž ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘”β€˜π‘–) = 𝑁)})
18 feq1 6698 . . . . . . . . . . . . . . . 16 (𝑔 = π‘Ž β†’ (𝑔:(1...(𝐾 + 1))βŸΆβ„•0 ↔ π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0))
19 simpl 483 . . . . . . . . . . . . . . . . . . 19 ((𝑔 = π‘Ž ∧ 𝑖 ∈ (1...(𝐾 + 1))) β†’ 𝑔 = π‘Ž)
2019fveq1d 6893 . . . . . . . . . . . . . . . . . 18 ((𝑔 = π‘Ž ∧ 𝑖 ∈ (1...(𝐾 + 1))) β†’ (π‘”β€˜π‘–) = (π‘Žβ€˜π‘–))
2120sumeq2dv 15651 . . . . . . . . . . . . . . . . 17 (𝑔 = π‘Ž β†’ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘”β€˜π‘–) = Σ𝑖 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘–))
2221eqeq1d 2734 . . . . . . . . . . . . . . . 16 (𝑔 = π‘Ž β†’ (Σ𝑖 ∈ (1...(𝐾 + 1))(π‘”β€˜π‘–) = 𝑁 ↔ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘–) = 𝑁))
2318, 22anbi12d 631 . . . . . . . . . . . . . . 15 (𝑔 = π‘Ž β†’ ((𝑔:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘”β€˜π‘–) = 𝑁) ↔ (π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘–) = 𝑁)))
2423elabg 3666 . . . . . . . . . . . . . 14 (π‘Ž ∈ 𝐴 β†’ (π‘Ž ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘”β€˜π‘–) = 𝑁)} ↔ (π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘–) = 𝑁)))
2513, 24syl 17 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (π‘Ž ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘”β€˜π‘–) = 𝑁)} ↔ (π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘–) = 𝑁)))
2625biimpd 228 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (π‘Ž ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘”β€˜π‘–) = 𝑁)} β†’ (π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘–) = 𝑁)))
2717, 26mpd 15 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘–) = 𝑁))
2827simpld 495 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0)
29283adant3 1132 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0)
30 eqid 2732 . . . . . . . . 9 (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(π‘Žβ€˜π‘™))) = (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(π‘Žβ€˜π‘™)))
31 fveq2 6891 . . . . . . . . . . . 12 (𝑖 = 𝑙 β†’ (π‘Žβ€˜π‘–) = (π‘Žβ€˜π‘™))
32 nfcv 2903 . . . . . . . . . . . 12 Ⅎ𝑙(1...(𝐾 + 1))
33 nfcv 2903 . . . . . . . . . . . 12 Ⅎ𝑖(1...(𝐾 + 1))
34 nfcv 2903 . . . . . . . . . . . 12 Ⅎ𝑙(π‘Žβ€˜π‘–)
35 nfcv 2903 . . . . . . . . . . . 12 Ⅎ𝑖(π‘Žβ€˜π‘™)
3631, 32, 33, 34, 35cbvsum 15643 . . . . . . . . . . 11 Σ𝑖 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘–) = Σ𝑙 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘™)
3727simprd 496 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘–) = 𝑁)
3836, 37eqtr3id 2786 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ Σ𝑙 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘™) = 𝑁)
39383adant3 1132 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ Σ𝑙 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘™) = 𝑁)
4010, 12, 29, 6, 30, 39sticksstones7 41060 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ ((𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(π‘Žβ€˜π‘™)))β€˜π‘—) ∈ (1...(𝑁 + 𝐾)))
418, 40eqeltrrd 2834 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)) ∈ (1...(𝑁 + 𝐾)))
42413expa 1118 . . . . . 6 (((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ 𝑗 ∈ (1...𝐾)) β†’ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)) ∈ (1...(𝑁 + 𝐾)))
43 eqid 2732 . . . . . 6 (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))) = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))
4442, 43fmptd 7115 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))):(1...𝐾)⟢(1...(𝑁 + 𝐾)))
459ad3antrrr 728 . . . . . . . . . 10 ((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑁 ∈ β„•0)
4645adantr 481 . . . . . . . . 9 (((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘₯ < 𝑦) β†’ 𝑁 ∈ β„•0)
4711ad3antrrr 728 . . . . . . . . . 10 ((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝐾 ∈ β„•0)
4847adantr 481 . . . . . . . . 9 (((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘₯ < 𝑦) β†’ 𝐾 ∈ β„•0)
4924adantl 482 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (π‘Ž ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘”β€˜π‘–) = 𝑁)} ↔ (π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘–) = 𝑁)))
5049biimpd 228 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (π‘Ž ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘”β€˜π‘–) = 𝑁)} β†’ (π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘–) = 𝑁)))
5117, 50mpd 15 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘–) = 𝑁))
5251simpld 495 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0)
5352adantr 481 . . . . . . . . . . 11 (((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) β†’ π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0)
5453adantr 481 . . . . . . . . . 10 ((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) β†’ π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0)
5554adantr 481 . . . . . . . . 9 (((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘₯ < 𝑦) β†’ π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0)
56 simpllr 774 . . . . . . . . 9 (((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘₯ < 𝑦) β†’ π‘₯ ∈ (1...𝐾))
57 simplr 767 . . . . . . . . 9 (((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘₯ < 𝑦) β†’ 𝑦 ∈ (1...𝐾))
58 simpr 485 . . . . . . . . 9 (((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘₯ < 𝑦) β†’ π‘₯ < 𝑦)
5946, 48, 55, 56, 57, 58, 43sticksstones6 41059 . . . . . . . 8 (((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘₯ < 𝑦) β†’ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘₯) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘¦))
6059ex 413 . . . . . . 7 ((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) β†’ (π‘₯ < 𝑦 β†’ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘₯) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘¦)))
6160ralrimiva 3146 . . . . . 6 (((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) β†’ βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘₯) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘¦)))
6261ralrimiva 3146 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘₯) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘¦)))
6344, 62jca 512 . . . 4 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))):(1...𝐾)⟢(1...(𝑁 + 𝐾)) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘₯) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘¦))))
64 fzfid 13940 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (1...𝐾) ∈ Fin)
6544, 64fexd 7231 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))) ∈ V)
66 feq1 6698 . . . . . . 7 (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))) β†’ (𝑓:(1...𝐾)⟢(1...(𝑁 + 𝐾)) ↔ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))):(1...𝐾)⟢(1...(𝑁 + 𝐾))))
67 fveq1 6890 . . . . . . . . . 10 (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))) β†’ (π‘“β€˜π‘₯) = ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘₯))
68 fveq1 6890 . . . . . . . . . 10 (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))) β†’ (π‘“β€˜π‘¦) = ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘¦))
6967, 68breq12d 5161 . . . . . . . . 9 (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))) β†’ ((π‘“β€˜π‘₯) < (π‘“β€˜π‘¦) ↔ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘₯) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘¦)))
7069imbi2d 340 . . . . . . . 8 (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))) β†’ ((π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)) ↔ (π‘₯ < 𝑦 β†’ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘₯) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘¦))))
71702ralbidv 3218 . . . . . . 7 (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))) β†’ (βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘₯) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘¦))))
7266, 71anbi12d 631 . . . . . 6 (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))) β†’ ((𝑓:(1...𝐾)⟢(1...(𝑁 + 𝐾)) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦))) ↔ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))):(1...𝐾)⟢(1...(𝑁 + 𝐾)) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘₯) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘¦)))))
7372elabg 3666 . . . . 5 ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))) ∈ V β†’ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))) ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟢(1...(𝑁 + 𝐾)) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)))} ↔ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))):(1...𝐾)⟢(1...(𝑁 + 𝐾)) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘₯) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘¦)))))
7465, 73syl 17 . . . 4 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))) ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟢(1...(𝑁 + 𝐾)) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)))} ↔ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))):(1...𝐾)⟢(1...(𝑁 + 𝐾)) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘₯) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))β€˜π‘¦)))))
7563, 74mpbird 256 . . 3 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))) ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟢(1...(𝑁 + 𝐾)) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)))})
76 sticksstones8.5 . . . 4 𝐡 = {𝑓 ∣ (𝑓:(1...𝐾)⟢(1...(𝑁 + 𝐾)) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)))}
7776a1i 11 . . 3 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ 𝐡 = {𝑓 ∣ (𝑓:(1...𝐾)⟢(1...(𝑁 + 𝐾)) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)))})
7875, 77eleqtrrd 2836 . 2 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))) ∈ 𝐡)
79 sticksstones8.3 . 2 𝐹 = (π‘Ž ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))))
8078, 79fmptd 7115 1 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  Vcvv 3474   class class class wbr 5148   ↦ cmpt 5231  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  Fincfn 8941  1c1 11113   + caddc 11115   < clt 11250  β„•0cn0 12474  ...cfz 13486  Ξ£csu 15634
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-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  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  ax-pre-sup 11190
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-oi 9507  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-z 12561  df-uz 12825  df-rp 12977  df-ico 13332  df-fz 13487  df-fzo 13630  df-seq 13969  df-exp 14030  df-hash 14293  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-clim 15434  df-sum 15635
This theorem is referenced by:  sticksstones11  41064  sticksstones12  41066
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