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Theorem sticksstones8 42135
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 (𝜑𝑁 ∈ ℕ0)
sticksstones8.2 (𝜑𝐾 ∈ ℕ0)
sticksstones8.3 𝐹 = (𝑎𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))))
sticksstones8.4 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)}
sticksstones8.5 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}
Assertion
Ref Expression
sticksstones8 (𝜑𝐹:𝐴𝐵)
Distinct variable groups:   𝐴,𝑎,𝑗,𝑙,𝑥,𝑦   𝐵,𝑎   𝑗,𝐾,𝑙,𝑓,𝑥,𝑦   𝑔,𝐾,𝑖   𝑓,𝑁,𝑗   𝑔,𝑁   𝑓,𝑎   𝑔,𝑎,𝑖   𝜑,𝑎,𝑗,𝑙   𝑖,𝑙   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑓,𝑔,𝑖)   𝐴(𝑓,𝑔,𝑖)   𝐵(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑙)   𝐹(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑎,𝑙)   𝐾(𝑎)   𝑁(𝑥,𝑦,𝑖,𝑎,𝑙)

Proof of Theorem sticksstones8
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2736 . . . . . . . . 9 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎𝑙))) = (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎𝑙))))
2 simpr 484 . . . . . . . . . 10 (((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) → 𝑒 = 𝑗)
32oveq2d 7447 . . . . . . . . . . 11 (((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) → (1...𝑒) = (1...𝑗))
43sumeq1d 15733 . . . . . . . . . 10 (((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) → Σ𝑙 ∈ (1...𝑒)(𝑎𝑙) = Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))
52, 4oveq12d 7449 . . . . . . . . 9 (((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) → (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎𝑙)) = (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))
6 simp3 1137 . . . . . . . . 9 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → 𝑗 ∈ (1...𝐾))
7 ovexd 7466 . . . . . . . . 9 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)) ∈ V)
81, 5, 6, 7fvmptd 7023 . . . . . . . 8 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → ((𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎𝑙)))‘𝑗) = (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))
9 sticksstones8.1 . . . . . . . . . 10 (𝜑𝑁 ∈ ℕ0)
1093ad2ant1 1132 . . . . . . . . 9 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → 𝑁 ∈ ℕ0)
11 sticksstones8.2 . . . . . . . . . 10 (𝜑𝐾 ∈ ℕ0)
12113ad2ant1 1132 . . . . . . . . 9 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → 𝐾 ∈ ℕ0)
13 simpr 484 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴) → 𝑎𝐴)
14 sticksstones8.4 . . . . . . . . . . . . . . 15 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)}
1514a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴) → 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)})
1615eqcomd 2741 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴) → {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)} = 𝐴)
1713, 16eleqtrrd 2842 . . . . . . . . . . . 12 ((𝜑𝑎𝐴) → 𝑎 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)})
18 feq1 6717 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑎 → (𝑔:(1...(𝐾 + 1))⟶ℕ0𝑎:(1...(𝐾 + 1))⟶ℕ0))
19 simpl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑔 = 𝑎𝑖 ∈ (1...(𝐾 + 1))) → 𝑔 = 𝑎)
2019fveq1d 6909 . . . . . . . . . . . . . . . . . 18 ((𝑔 = 𝑎𝑖 ∈ (1...(𝐾 + 1))) → (𝑔𝑖) = (𝑎𝑖))
2120sumeq2dv 15735 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑎 → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎𝑖))
2221eqeq1d 2737 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑎 → (Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎𝑖) = 𝑁))
2318, 22anbi12d 632 . . . . . . . . . . . . . . 15 (𝑔 = 𝑎 → ((𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁) ↔ (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎𝑖) = 𝑁)))
2423elabg 3677 . . . . . . . . . . . . . 14 (𝑎𝐴 → (𝑎 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)} ↔ (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎𝑖) = 𝑁)))
2513, 24syl 17 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴) → (𝑎 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)} ↔ (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎𝑖) = 𝑁)))
2625biimpd 229 . . . . . . . . . . . 12 ((𝜑𝑎𝐴) → (𝑎 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)} → (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎𝑖) = 𝑁)))
2717, 26mpd 15 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎𝑖) = 𝑁))
2827simpld 494 . . . . . . . . . 10 ((𝜑𝑎𝐴) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
29283adant3 1131 . . . . . . . . 9 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
30 eqid 2735 . . . . . . . . 9 (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎𝑙))) = (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎𝑙)))
31 fveq2 6907 . . . . . . . . . . . 12 (𝑖 = 𝑙 → (𝑎𝑖) = (𝑎𝑙))
32 nfcv 2903 . . . . . . . . . . . 12 𝑙(𝑎𝑖)
33 nfcv 2903 . . . . . . . . . . . 12 𝑖(𝑎𝑙)
3431, 32, 33cbvsum 15728 . . . . . . . . . . 11 Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎𝑖) = Σ𝑙 ∈ (1...(𝐾 + 1))(𝑎𝑙)
3527simprd 495 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎𝑖) = 𝑁)
3634, 35eqtr3id 2789 . . . . . . . . . 10 ((𝜑𝑎𝐴) → Σ𝑙 ∈ (1...(𝐾 + 1))(𝑎𝑙) = 𝑁)
37363adant3 1131 . . . . . . . . 9 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → Σ𝑙 ∈ (1...(𝐾 + 1))(𝑎𝑙) = 𝑁)
3810, 12, 29, 6, 30, 37sticksstones7 42134 . . . . . . . 8 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → ((𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎𝑙)))‘𝑗) ∈ (1...(𝑁 + 𝐾)))
398, 38eqeltrrd 2840 . . . . . . 7 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)) ∈ (1...(𝑁 + 𝐾)))
40393expa 1117 . . . . . 6 (((𝜑𝑎𝐴) ∧ 𝑗 ∈ (1...𝐾)) → (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)) ∈ (1...(𝑁 + 𝐾)))
41 eqid 2735 . . . . . 6 (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))
4240, 41fmptd 7134 . . . . 5 ((𝜑𝑎𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))):(1...𝐾)⟶(1...(𝑁 + 𝐾)))
439ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) → 𝑁 ∈ ℕ0)
4443adantr 480 . . . . . . . . 9 (((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝑁 ∈ ℕ0)
4511ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) → 𝐾 ∈ ℕ0)
4645adantr 480 . . . . . . . . 9 (((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝐾 ∈ ℕ0)
4724adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑎𝐴) → (𝑎 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)} ↔ (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎𝑖) = 𝑁)))
4847biimpd 229 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴) → (𝑎 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)} → (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎𝑖) = 𝑁)))
4917, 48mpd 15 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴) → (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎𝑖) = 𝑁))
5049simpld 494 . . . . . . . . . . . 12 ((𝜑𝑎𝐴) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
5150adantr 480 . . . . . . . . . . 11 (((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
5251adantr 480 . . . . . . . . . 10 ((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
5352adantr 480 . . . . . . . . 9 (((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
54 simpllr 776 . . . . . . . . 9 (((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝑥 ∈ (1...𝐾))
55 simplr 769 . . . . . . . . 9 (((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (1...𝐾))
56 simpr 484 . . . . . . . . 9 (((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
5744, 46, 53, 54, 55, 56, 41sticksstones6 42133 . . . . . . . 8 (((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑦))
5857ex 412 . . . . . . 7 ((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) → (𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑦)))
5958ralrimiva 3144 . . . . . 6 (((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) → ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑦)))
6059ralrimiva 3144 . . . . 5 ((𝜑𝑎𝐴) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑦)))
6142, 60jca 511 . . . 4 ((𝜑𝑎𝐴) → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))):(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑦))))
62 fzfid 14011 . . . . . 6 ((𝜑𝑎𝐴) → (1...𝐾) ∈ Fin)
6342, 62fexd 7247 . . . . 5 ((𝜑𝑎𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) ∈ V)
64 feq1 6717 . . . . . . 7 (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) → (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ↔ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))):(1...𝐾)⟶(1...(𝑁 + 𝐾))))
65 fveq1 6906 . . . . . . . . . 10 (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) → (𝑓𝑥) = ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑥))
66 fveq1 6906 . . . . . . . . . 10 (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) → (𝑓𝑦) = ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑦))
6765, 66breq12d 5161 . . . . . . . . 9 (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) → ((𝑓𝑥) < (𝑓𝑦) ↔ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑦)))
6867imbi2d 340 . . . . . . . 8 (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) → ((𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ (𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑦))))
69682ralbidv 3219 . . . . . . 7 (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑦))))
7064, 69anbi12d 632 . . . . . 6 (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) → ((𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))) ↔ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))):(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑦)))))
7170elabg 3677 . . . . 5 ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) ∈ V → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))} ↔ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))):(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑦)))))
7263, 71syl 17 . . . 4 ((𝜑𝑎𝐴) → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))} ↔ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))):(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑦)))))
7361, 72mpbird 257 . . 3 ((𝜑𝑎𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))})
74 sticksstones8.5 . . . 4 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}
7574a1i 11 . . 3 ((𝜑𝑎𝐴) → 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))})
7673, 75eleqtrrd 2842 . 2 ((𝜑𝑎𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) ∈ 𝐵)
77 sticksstones8.3 . 2 𝐹 = (𝑎𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))))
7876, 77fmptd 7134 1 (𝜑𝐹:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wral 3059  Vcvv 3478   class class class wbr 5148  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  Fincfn 8984  1c1 11154   + caddc 11156   < clt 11293  0cn0 12524  ...cfz 13544  Σcsu 15719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-ico 13390  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720
This theorem is referenced by:  sticksstones11  42138  sticksstones12  42140
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