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Theorem sticksstones8 40969
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 2734 . . . . . . . . 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 486 . . . . . . . . . 10 (((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) β†’ 𝑒 = 𝑗)
32oveq2d 7425 . . . . . . . . . . 11 (((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) β†’ (1...𝑒) = (1...𝑗))
43sumeq1d 15647 . . . . . . . . . 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 7427 . . . . . . . . 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 1139 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ 𝑗 ∈ (1...𝐾))
7 ovexd 7444 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)) ∈ V)
81, 5, 6, 7fvmptd 7006 . . . . . . . 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 1134 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ 𝑁 ∈ β„•0)
11 sticksstones8.2 . . . . . . . . . 10 (πœ‘ β†’ 𝐾 ∈ β„•0)
12113ad2ant1 1134 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ 𝐾 ∈ β„•0)
13 simpr 486 . . . . . . . . . . . . 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 2739 . . . . . . . . . . . . 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 2837 . . . . . . . . . . . 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 6699 . . . . . . . . . . . . . . . 16 (𝑔 = π‘Ž β†’ (𝑔:(1...(𝐾 + 1))βŸΆβ„•0 ↔ π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0))
19 simpl 484 . . . . . . . . . . . . . . . . . . 19 ((𝑔 = π‘Ž ∧ 𝑖 ∈ (1...(𝐾 + 1))) β†’ 𝑔 = π‘Ž)
2019fveq1d 6894 . . . . . . . . . . . . . . . . . 18 ((𝑔 = π‘Ž ∧ 𝑖 ∈ (1...(𝐾 + 1))) β†’ (π‘”β€˜π‘–) = (π‘Žβ€˜π‘–))
2120sumeq2dv 15649 . . . . . . . . . . . . . . . . 17 (𝑔 = π‘Ž β†’ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘”β€˜π‘–) = Σ𝑖 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘–))
2221eqeq1d 2735 . . . . . . . . . . . . . . . 16 (𝑔 = π‘Ž β†’ (Σ𝑖 ∈ (1...(𝐾 + 1))(π‘”β€˜π‘–) = 𝑁 ↔ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘–) = 𝑁))
2318, 22anbi12d 632 . . . . . . . . . . . . . . 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 3667 . . . . . . . . . . . . . 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 496 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0)
29283adant3 1133 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0)
30 eqid 2733 . . . . . . . . 9 (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(π‘Žβ€˜π‘™))) = (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(π‘Žβ€˜π‘™)))
31 fveq2 6892 . . . . . . . . . . . 12 (𝑖 = 𝑙 β†’ (π‘Žβ€˜π‘–) = (π‘Žβ€˜π‘™))
32 nfcv 2904 . . . . . . . . . . . 12 Ⅎ𝑙(1...(𝐾 + 1))
33 nfcv 2904 . . . . . . . . . . . 12 Ⅎ𝑖(1...(𝐾 + 1))
34 nfcv 2904 . . . . . . . . . . . 12 Ⅎ𝑙(π‘Žβ€˜π‘–)
35 nfcv 2904 . . . . . . . . . . . 12 Ⅎ𝑖(π‘Žβ€˜π‘™)
3631, 32, 33, 34, 35cbvsum 15641 . . . . . . . . . . 11 Σ𝑖 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘–) = Σ𝑙 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘™)
3727simprd 497 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ Σ𝑖 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘–) = 𝑁)
3836, 37eqtr3id 2787 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ Σ𝑙 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘™) = 𝑁)
39383adant3 1133 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ Σ𝑙 ∈ (1...(𝐾 + 1))(π‘Žβ€˜π‘™) = 𝑁)
4010, 12, 29, 6, 30, 39sticksstones7 40968 . . . . . . . 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 2835 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) β†’ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)) ∈ (1...(𝑁 + 𝐾)))
42413expa 1119 . . . . . 6 (((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ 𝑗 ∈ (1...𝐾)) β†’ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)) ∈ (1...(𝑁 + 𝐾)))
43 eqid 2733 . . . . . 6 (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))) = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™)))
4442, 43fmptd 7114 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))):(1...𝐾)⟢(1...(𝑁 + 𝐾)))
459ad3antrrr 729 . . . . . . . . . 10 ((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑁 ∈ β„•0)
4645adantr 482 . . . . . . . . 9 (((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘₯ < 𝑦) β†’ 𝑁 ∈ β„•0)
4711ad3antrrr 729 . . . . . . . . . 10 ((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝐾 ∈ β„•0)
4847adantr 482 . . . . . . . . 9 (((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘₯ < 𝑦) β†’ 𝐾 ∈ β„•0)
4924adantl 483 . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0)
5352adantr 482 . . . . . . . . . . 11 (((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) β†’ π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0)
5453adantr 482 . . . . . . . . . 10 ((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) β†’ π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0)
5554adantr 482 . . . . . . . . 9 (((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘₯ < 𝑦) β†’ π‘Ž:(1...(𝐾 + 1))βŸΆβ„•0)
56 simpllr 775 . . . . . . . . 9 (((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘₯ < 𝑦) β†’ π‘₯ ∈ (1...𝐾))
57 simplr 768 . . . . . . . . 9 (((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘₯ < 𝑦) β†’ 𝑦 ∈ (1...𝐾))
58 simpr 486 . . . . . . . . 9 (((((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘₯ < 𝑦) β†’ π‘₯ < 𝑦)
5946, 48, 55, 56, 57, 58, 43sticksstones6 40967 . . . . . . . 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 414 . . . . . . 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 3147 . . . . . 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 3147 . . . . 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 513 . . . 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 13938 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (1...𝐾) ∈ Fin)
6544, 64fexd 7229 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(π‘Žβ€˜π‘™))) ∈ V)
66 feq1 6699 . . . . . . 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 6891 . . . . . . . . . 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 6891 . . . . . . . . . 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 5162 . . . . . . . . 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 341 . . . . . . . 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 3219 . . . . . . 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 632 . . . . . 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 3667 . . . . 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 257 . . 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 2837 . 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 7114 1 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3062  Vcvv 3475   class class class wbr 5149   ↦ cmpt 5232  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Fincfn 8939  1c1 11111   + caddc 11113   < clt 11248  β„•0cn0 12472  ...cfz 13484  Ξ£csu 15632
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-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-rp 12975  df-ico 13330  df-fz 13485  df-fzo 13628  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-sum 15633
This theorem is referenced by:  sticksstones11  40972  sticksstones12  40974
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