Users' Mathboxes Mathbox for metakunt < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sticksstones8 Structured version   Visualization version   GIF version

Theorem sticksstones8 40907
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 2734 . . . . . . . . 9 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎𝑙))) = (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎𝑙))))
2 simpr 486 . . . . . . . . . 10 (((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) → 𝑒 = 𝑗)
32oveq2d 7420 . . . . . . . . . . 11 (((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) → (1...𝑒) = (1...𝑗))
43sumeq1d 15643 . . . . . . . . . 10 (((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) → Σ𝑙 ∈ (1...𝑒)(𝑎𝑙) = Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))
52, 4oveq12d 7422 . . . . . . . . 9 (((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) → (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎𝑙)) = (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))
6 simp3 1139 . . . . . . . . 9 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → 𝑗 ∈ (1...𝐾))
7 ovexd 7439 . . . . . . . . 9 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)) ∈ V)
81, 5, 6, 7fvmptd 7001 . . . . . . . 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 1134 . . . . . . . . 9 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → 𝑁 ∈ ℕ0)
11 sticksstones8.2 . . . . . . . . . 10 (𝜑𝐾 ∈ ℕ0)
12113ad2ant1 1134 . . . . . . . . 9 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → 𝐾 ∈ ℕ0)
13 simpr 486 . . . . . . . . . . . . 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 2739 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴) → {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)} = 𝐴)
1713, 16eleqtrrd 2837 . . . . . . . . . . . 12 ((𝜑𝑎𝐴) → 𝑎 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)})
18 feq1 6695 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑎 → (𝑔:(1...(𝐾 + 1))⟶ℕ0𝑎:(1...(𝐾 + 1))⟶ℕ0))
19 simpl 484 . . . . . . . . . . . . . . . . . . 19 ((𝑔 = 𝑎𝑖 ∈ (1...(𝐾 + 1))) → 𝑔 = 𝑎)
2019fveq1d 6890 . . . . . . . . . . . . . . . . . 18 ((𝑔 = 𝑎𝑖 ∈ (1...(𝐾 + 1))) → (𝑔𝑖) = (𝑎𝑖))
2120sumeq2dv 15645 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑎 → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎𝑖))
2221eqeq1d 2735 . . . . . . . . . . . . . . . 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 3665 . . . . . . . . . . . . . 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 228 . . . . . . . . . . . 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 496 . . . . . . . . . 10 ((𝜑𝑎𝐴) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
29283adant3 1133 . . . . . . . . 9 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
30 eqid 2733 . . . . . . . . 9 (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎𝑙))) = (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎𝑙)))
31 fveq2 6888 . . . . . . . . . . . 12 (𝑖 = 𝑙 → (𝑎𝑖) = (𝑎𝑙))
32 nfcv 2904 . . . . . . . . . . . 12 𝑙(1...(𝐾 + 1))
33 nfcv 2904 . . . . . . . . . . . 12 𝑖(1...(𝐾 + 1))
34 nfcv 2904 . . . . . . . . . . . 12 𝑙(𝑎𝑖)
35 nfcv 2904 . . . . . . . . . . . 12 𝑖(𝑎𝑙)
3631, 32, 33, 34, 35cbvsum 15637 . . . . . . . . . . 11 Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎𝑖) = Σ𝑙 ∈ (1...(𝐾 + 1))(𝑎𝑙)
3727simprd 497 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎𝑖) = 𝑁)
3836, 37eqtr3id 2787 . . . . . . . . . 10 ((𝜑𝑎𝐴) → Σ𝑙 ∈ (1...(𝐾 + 1))(𝑎𝑙) = 𝑁)
39383adant3 1133 . . . . . . . . 9 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → Σ𝑙 ∈ (1...(𝐾 + 1))(𝑎𝑙) = 𝑁)
4010, 12, 29, 6, 30, 39sticksstones7 40906 . . . . . . . 8 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → ((𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎𝑙)))‘𝑗) ∈ (1...(𝑁 + 𝐾)))
418, 40eqeltrrd 2835 . . . . . . 7 ((𝜑𝑎𝐴𝑗 ∈ (1...𝐾)) → (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)) ∈ (1...(𝑁 + 𝐾)))
42413expa 1119 . . . . . 6 (((𝜑𝑎𝐴) ∧ 𝑗 ∈ (1...𝐾)) → (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)) ∈ (1...(𝑁 + 𝐾)))
43 eqid 2733 . . . . . 6 (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))
4442, 43fmptd 7109 . . . . 5 ((𝜑𝑎𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))):(1...𝐾)⟶(1...(𝑁 + 𝐾)))
459ad3antrrr 729 . . . . . . . . . 10 ((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) → 𝑁 ∈ ℕ0)
4645adantr 482 . . . . . . . . 9 (((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝑁 ∈ ℕ0)
4711ad3antrrr 729 . . . . . . . . . 10 ((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) → 𝐾 ∈ ℕ0)
4847adantr 482 . . . . . . . . 9 (((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝐾 ∈ ℕ0)
4924adantl 483 . . . . . . . . . . . . . . 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)))
5049biimpd 228 . . . . . . . . . . . . . 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)))
5117, 50mpd 15 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴) → (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎𝑖) = 𝑁))
5251simpld 496 . . . . . . . . . . . 12 ((𝜑𝑎𝐴) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
5352adantr 482 . . . . . . . . . . 11 (((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
5453adantr 482 . . . . . . . . . 10 ((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
5554adantr 482 . . . . . . . . 9 (((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
56 simpllr 775 . . . . . . . . 9 (((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝑥 ∈ (1...𝐾))
57 simplr 768 . . . . . . . . 9 (((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (1...𝐾))
58 simpr 486 . . . . . . . . 9 (((((𝜑𝑎𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
5946, 48, 55, 56, 57, 58, 43sticksstones6 40905 . . . . . . . 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))
6059ex 414 . . . . . . 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)))
6160ralrimiva 3147 . . . . . 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)))
6261ralrimiva 3147 . . . . 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)))
6344, 62jca 513 . . . 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))))
64 fzfid 13934 . . . . . 6 ((𝜑𝑎𝐴) → (1...𝐾) ∈ Fin)
6544, 64fexd 7224 . . . . 5 ((𝜑𝑎𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) ∈ V)
66 feq1 6695 . . . . . . 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))))
67 fveq1 6887 . . . . . . . . . 10 (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) → (𝑓𝑥) = ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑥))
68 fveq1 6887 . . . . . . . . . 10 (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) → (𝑓𝑦) = ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙)))‘𝑦))
6967, 68breq12d 5160 . . . . . . . . 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)))
7069imbi2d 341 . . . . . . . 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))))
71702ralbidv 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))))
7266, 71anbi12d 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)))))
7372elabg 3665 . . . . 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)))))
7465, 73syl 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)))))
7563, 74mpbird 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)))})
76 sticksstones8.5 . . . 4 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}
7776a1i 11 . . 3 ((𝜑𝑎𝐴) → 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))})
7875, 77eleqtrrd 2837 . 2 ((𝜑𝑎𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))) ∈ 𝐵)
79 sticksstones8.3 . 2 𝐹 = (𝑎𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))))
8078, 79fmptd 7109 1 (𝜑𝐹:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wral 3062  Vcvv 3475   class class class wbr 5147  cmpt 5230  wf 6536  cfv 6540  (class class class)co 7404  Fincfn 8935  1c1 11107   + caddc 11109   < clt 11244  0cn0 12468  ...cfz 13480  Σcsu 15628
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-ico 13326  df-fz 13481  df-fzo 13624  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-sum 15629
This theorem is referenced by:  sticksstones11  40910  sticksstones12  40912
  Copyright terms: Public domain W3C validator