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Theorem sticksstones18 39872
Description: Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)
Hypotheses
Ref Expression
sticksstones18.1 (𝜑𝑁 ∈ ℕ0)
sticksstones18.2 (𝜑𝐾 ∈ ℕ0)
sticksstones18.3 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)}
sticksstones18.4 𝐵 = { ∣ (:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁)}
sticksstones18.5 (𝜑𝑍:(1...𝐾)–1-1-onto𝑆)
sticksstones18.6 𝐹 = (𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))
Assertion
Ref Expression
sticksstones18 (𝜑𝐹:𝐴𝐵)
Distinct variable groups:   𝐴,𝑎,𝑖,𝑥   𝐵,𝑎   𝑔,𝐾,𝑖   𝑔,𝑁   ,𝑁   𝑆,,𝑖,𝑥   ,𝑍,𝑖,𝑥   𝑔,𝑎   ,𝑎   𝜑,𝑎,𝑖,𝑥
Allowed substitution hints:   𝜑(𝑔,)   𝐴(𝑔,)   𝐵(𝑥,𝑔,,𝑖)   𝑆(𝑔,𝑎)   𝐹(𝑥,𝑔,,𝑖,𝑎)   𝐾(𝑥,,𝑎)   𝑁(𝑥,𝑖,𝑎)   𝑍(𝑔,𝑎)

Proof of Theorem sticksstones18
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 sticksstones18.3 . . . . . . . . . . . . . 14 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)}
21eqimssi 3974 . . . . . . . . . . . . 13 𝐴 ⊆ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)}
32a1i 11 . . . . . . . . . . . 12 (𝜑𝐴 ⊆ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)})
43sseld 3915 . . . . . . . . . . 11 (𝜑 → (𝑎𝐴𝑎 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)}))
54imp 410 . . . . . . . . . 10 ((𝜑𝑎𝐴) → 𝑎 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)})
6 vex 3425 . . . . . . . . . . 11 𝑎 ∈ V
7 feq1 6545 . . . . . . . . . . . 12 (𝑔 = 𝑎 → (𝑔:(1...𝐾)⟶ℕ0𝑎:(1...𝐾)⟶ℕ0))
8 simpl 486 . . . . . . . . . . . . . . 15 ((𝑔 = 𝑎𝑖 ∈ (1...𝐾)) → 𝑔 = 𝑎)
98fveq1d 6738 . . . . . . . . . . . . . 14 ((𝑔 = 𝑎𝑖 ∈ (1...𝐾)) → (𝑔𝑖) = (𝑎𝑖))
109sumeq2dv 15292 . . . . . . . . . . . . 13 (𝑔 = 𝑎 → Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = Σ𝑖 ∈ (1...𝐾)(𝑎𝑖))
1110eqeq1d 2740 . . . . . . . . . . . 12 (𝑔 = 𝑎 → (Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...𝐾)(𝑎𝑖) = 𝑁))
127, 11anbi12d 634 . . . . . . . . . . 11 (𝑔 = 𝑎 → ((𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁) ↔ (𝑎:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑎𝑖) = 𝑁)))
136, 12elab 3600 . . . . . . . . . 10 (𝑎 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)} ↔ (𝑎:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑎𝑖) = 𝑁))
145, 13sylib 221 . . . . . . . . 9 ((𝜑𝑎𝐴) → (𝑎:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑎𝑖) = 𝑁))
1514simpld 498 . . . . . . . 8 ((𝜑𝑎𝐴) → 𝑎:(1...𝐾)⟶ℕ0)
1615adantr 484 . . . . . . 7 (((𝜑𝑎𝐴) ∧ 𝑥𝑆) → 𝑎:(1...𝐾)⟶ℕ0)
17 sticksstones18.5 . . . . . . . . . . 11 (𝜑𝑍:(1...𝐾)–1-1-onto𝑆)
18 f1ocnv 6692 . . . . . . . . . . 11 (𝑍:(1...𝐾)–1-1-onto𝑆𝑍:𝑆1-1-onto→(1...𝐾))
1917, 18syl 17 . . . . . . . . . 10 (𝜑𝑍:𝑆1-1-onto→(1...𝐾))
20 f1of 6680 . . . . . . . . . 10 (𝑍:𝑆1-1-onto→(1...𝐾) → 𝑍:𝑆⟶(1...𝐾))
2119, 20syl 17 . . . . . . . . 9 (𝜑𝑍:𝑆⟶(1...𝐾))
2221adantr 484 . . . . . . . 8 ((𝜑𝑎𝐴) → 𝑍:𝑆⟶(1...𝐾))
2322ffvelrnda 6923 . . . . . . 7 (((𝜑𝑎𝐴) ∧ 𝑥𝑆) → (𝑍𝑥) ∈ (1...𝐾))
2416, 23ffvelrnd 6924 . . . . . 6 (((𝜑𝑎𝐴) ∧ 𝑥𝑆) → (𝑎‘(𝑍𝑥)) ∈ ℕ0)
2524fmpttd 6951 . . . . 5 ((𝜑𝑎𝐴) → (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))):𝑆⟶ℕ0)
26 eqidd 2739 . . . . . . . 8 (((𝜑𝑎𝐴) ∧ 𝑖𝑆) → (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) = (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))
27 simpr 488 . . . . . . . . . 10 ((((𝜑𝑎𝐴) ∧ 𝑖𝑆) ∧ 𝑥 = 𝑖) → 𝑥 = 𝑖)
2827fveq2d 6740 . . . . . . . . 9 ((((𝜑𝑎𝐴) ∧ 𝑖𝑆) ∧ 𝑥 = 𝑖) → (𝑍𝑥) = (𝑍𝑖))
2928fveq2d 6740 . . . . . . . 8 ((((𝜑𝑎𝐴) ∧ 𝑖𝑆) ∧ 𝑥 = 𝑖) → (𝑎‘(𝑍𝑥)) = (𝑎‘(𝑍𝑖)))
30 simpr 488 . . . . . . . 8 (((𝜑𝑎𝐴) ∧ 𝑖𝑆) → 𝑖𝑆)
31 fvexd 6751 . . . . . . . 8 (((𝜑𝑎𝐴) ∧ 𝑖𝑆) → (𝑎‘(𝑍𝑖)) ∈ V)
3226, 29, 30, 31fvmptd 6844 . . . . . . 7 (((𝜑𝑎𝐴) ∧ 𝑖𝑆) → ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖) = (𝑎‘(𝑍𝑖)))
3332sumeq2dv 15292 . . . . . 6 ((𝜑𝑎𝐴) → Σ𝑖𝑆 ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖) = Σ𝑖𝑆 (𝑎‘(𝑍𝑖)))
34 fveq2 6736 . . . . . . . . 9 (𝑛 = (𝑍𝑖) → (𝑎𝑛) = (𝑎‘(𝑍𝑖)))
35 fzfid 13571 . . . . . . . . . 10 ((𝜑𝑎𝐴) → (1...𝐾) ∈ Fin)
3617adantr 484 . . . . . . . . . . . 12 ((𝜑𝑎𝐴) → 𝑍:(1...𝐾)–1-1-onto𝑆)
37 f1oenfi 8882 . . . . . . . . . . . 12 (((1...𝐾) ∈ Fin ∧ 𝑍:(1...𝐾)–1-1-onto𝑆) → (1...𝐾) ≈ 𝑆)
3835, 36, 37syl2anc 587 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → (1...𝐾) ≈ 𝑆)
3938ensymd 8702 . . . . . . . . . 10 ((𝜑𝑎𝐴) → 𝑆 ≈ (1...𝐾))
40 enfii 8888 . . . . . . . . . 10 (((1...𝐾) ∈ Fin ∧ 𝑆 ≈ (1...𝐾)) → 𝑆 ∈ Fin)
4135, 39, 40syl2anc 587 . . . . . . . . 9 ((𝜑𝑎𝐴) → 𝑆 ∈ Fin)
4219adantr 484 . . . . . . . . 9 ((𝜑𝑎𝐴) → 𝑍:𝑆1-1-onto→(1...𝐾))
43 eqidd 2739 . . . . . . . . 9 (((𝜑𝑎𝐴) ∧ 𝑖𝑆) → (𝑍𝑖) = (𝑍𝑖))
44 nn0sscn 12120 . . . . . . . . . . . 12 0 ⊆ ℂ
4544a1i 11 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → ℕ0 ⊆ ℂ)
46 fss 6581 . . . . . . . . . . 11 ((𝑎:(1...𝐾)⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑎:(1...𝐾)⟶ℂ)
4715, 45, 46syl2anc 587 . . . . . . . . . 10 ((𝜑𝑎𝐴) → 𝑎:(1...𝐾)⟶ℂ)
4847ffvelrnda 6923 . . . . . . . . 9 (((𝜑𝑎𝐴) ∧ 𝑛 ∈ (1...𝐾)) → (𝑎𝑛) ∈ ℂ)
4934, 41, 42, 43, 48fsumf1o 15312 . . . . . . . 8 ((𝜑𝑎𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎𝑛) = Σ𝑖𝑆 (𝑎‘(𝑍𝑖)))
5049eqcomd 2744 . . . . . . 7 ((𝜑𝑎𝐴) → Σ𝑖𝑆 (𝑎‘(𝑍𝑖)) = Σ𝑛 ∈ (1...𝐾)(𝑎𝑛))
51 fveq2 6736 . . . . . . . . . 10 (𝑛 = 𝑖 → (𝑎𝑛) = (𝑎𝑖))
5251cbvsumv 15285 . . . . . . . . 9 Σ𝑛 ∈ (1...𝐾)(𝑎𝑛) = Σ𝑖 ∈ (1...𝐾)(𝑎𝑖)
5352a1i 11 . . . . . . . 8 ((𝜑𝑎𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎𝑛) = Σ𝑖 ∈ (1...𝐾)(𝑎𝑖))
5414simprd 499 . . . . . . . 8 ((𝜑𝑎𝐴) → Σ𝑖 ∈ (1...𝐾)(𝑎𝑖) = 𝑁)
5553, 54eqtrd 2778 . . . . . . 7 ((𝜑𝑎𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎𝑛) = 𝑁)
5650, 55eqtrd 2778 . . . . . 6 ((𝜑𝑎𝐴) → Σ𝑖𝑆 (𝑎‘(𝑍𝑖)) = 𝑁)
5733, 56eqtrd 2778 . . . . 5 ((𝜑𝑎𝐴) → Σ𝑖𝑆 ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖) = 𝑁)
5825, 57jca 515 . . . 4 ((𝜑𝑎𝐴) → ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))):𝑆⟶ℕ0 ∧ Σ𝑖𝑆 ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖) = 𝑁))
59 fzfid 13571 . . . . . . . 8 (𝜑 → (1...𝐾) ∈ Fin)
6059adantr 484 . . . . . . 7 ((𝜑𝑎𝐴) → (1...𝐾) ∈ Fin)
6159, 17, 37syl2anc 587 . . . . . . . . 9 (𝜑 → (1...𝐾) ≈ 𝑆)
6261ensymd 8702 . . . . . . . 8 (𝜑𝑆 ≈ (1...𝐾))
6362adantr 484 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑆 ≈ (1...𝐾))
6460, 63, 40syl2anc 587 . . . . . 6 ((𝜑𝑎𝐴) → 𝑆 ∈ Fin)
6564mptexd 7059 . . . . 5 ((𝜑𝑎𝐴) → (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) ∈ V)
66 feq1 6545 . . . . . . 7 ( = (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) → (:𝑆⟶ℕ0 ↔ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))):𝑆⟶ℕ0))
67 simpl 486 . . . . . . . . . 10 (( = (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) ∧ 𝑖𝑆) → = (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))
6867fveq1d 6738 . . . . . . . . 9 (( = (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) ∧ 𝑖𝑆) → (𝑖) = ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖))
6968sumeq2dv 15292 . . . . . . . 8 ( = (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) → Σ𝑖𝑆 (𝑖) = Σ𝑖𝑆 ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖))
7069eqeq1d 2740 . . . . . . 7 ( = (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) → (Σ𝑖𝑆 (𝑖) = 𝑁 ↔ Σ𝑖𝑆 ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖) = 𝑁))
7166, 70anbi12d 634 . . . . . 6 ( = (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) → ((:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁) ↔ ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))):𝑆⟶ℕ0 ∧ Σ𝑖𝑆 ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖) = 𝑁)))
7271elabg 3598 . . . . 5 ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) ∈ V → ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) ∈ { ∣ (:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁)} ↔ ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))):𝑆⟶ℕ0 ∧ Σ𝑖𝑆 ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖) = 𝑁)))
7365, 72syl 17 . . . 4 ((𝜑𝑎𝐴) → ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) ∈ { ∣ (:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁)} ↔ ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))):𝑆⟶ℕ0 ∧ Σ𝑖𝑆 ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖) = 𝑁)))
7458, 73mpbird 260 . . 3 ((𝜑𝑎𝐴) → (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) ∈ { ∣ (:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁)})
75 sticksstones18.4 . . . 4 𝐵 = { ∣ (:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁)}
7675a1i 11 . . 3 ((𝜑𝑎𝐴) → 𝐵 = { ∣ (:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁)})
7774, 76eleqtrrd 2842 . 2 ((𝜑𝑎𝐴) → (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) ∈ 𝐵)
78 sticksstones18.6 . 2 𝐹 = (𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))
7977, 78fmptd 6950 1 (𝜑𝐹:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2111  {cab 2715  Vcvv 3421  wss 3881   class class class wbr 5068  cmpt 5150  ccnv 5565  wf 6394  1-1-ontowf1o 6397  cfv 6398  (class class class)co 7232  cen 8644  Fincfn 8647  cc 10752  1c1 10755  0cn0 12115  ...cfz 13120  Σcsu 15274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5194  ax-sep 5207  ax-nul 5214  ax-pow 5273  ax-pr 5337  ax-un 7542  ax-inf2 9281  ax-cnex 10810  ax-resscn 10811  ax-1cn 10812  ax-icn 10813  ax-addcl 10814  ax-addrcl 10815  ax-mulcl 10816  ax-mulrcl 10817  ax-mulcom 10818  ax-addass 10819  ax-mulass 10820  ax-distr 10821  ax-i2m1 10822  ax-1ne0 10823  ax-1rid 10824  ax-rnegex 10825  ax-rrecex 10826  ax-cnre 10827  ax-pre-lttri 10828  ax-pre-lttrn 10829  ax-pre-ltadd 10830  ax-pre-mulgt0 10831  ax-pre-sup 10832
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3067  df-rex 3068  df-reu 3069  df-rmo 3070  df-rab 3071  df-v 3423  df-sbc 3710  df-csb 3827  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4253  df-if 4455  df-pw 4530  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4835  df-int 4875  df-iun 4921  df-br 5069  df-opab 5131  df-mpt 5151  df-tr 5177  df-id 5470  df-eprel 5475  df-po 5483  df-so 5484  df-fr 5524  df-se 5525  df-we 5526  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-pred 6176  df-ord 6234  df-on 6235  df-lim 6236  df-suc 6237  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-f1 6403  df-fo 6404  df-f1o 6405  df-fv 6406  df-isom 6407  df-riota 7189  df-ov 7235  df-oprab 7236  df-mpo 7237  df-om 7664  df-1st 7780  df-2nd 7781  df-wrecs 8068  df-recs 8129  df-rdg 8167  df-1o 8223  df-er 8412  df-en 8648  df-dom 8649  df-sdom 8650  df-fin 8651  df-sup 9083  df-oi 9151  df-card 9580  df-pnf 10894  df-mnf 10895  df-xr 10896  df-ltxr 10897  df-le 10898  df-sub 11089  df-neg 11090  df-div 11515  df-nn 11856  df-2 11918  df-3 11919  df-n0 12116  df-z 12202  df-uz 12464  df-rp 12612  df-fz 13121  df-fzo 13264  df-seq 13600  df-exp 13661  df-hash 13922  df-cj 14687  df-re 14688  df-im 14689  df-sqrt 14823  df-abs 14824  df-clim 15074  df-sum 15275
This theorem is referenced by:  sticksstones19  39873
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