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Theorem sticksstones18 42166
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 4043 . . . . . . . . . . . . 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 3981 . . . . . . . . . . 11 (𝜑 → (𝑎𝐴𝑎 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)}))
54imp 406 . . . . . . . . . 10 ((𝜑𝑎𝐴) → 𝑎 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)})
6 vex 3483 . . . . . . . . . . 11 𝑎 ∈ V
7 feq1 6715 . . . . . . . . . . . 12 (𝑔 = 𝑎 → (𝑔:(1...𝐾)⟶ℕ0𝑎:(1...𝐾)⟶ℕ0))
8 simpl 482 . . . . . . . . . . . . . . 15 ((𝑔 = 𝑎𝑖 ∈ (1...𝐾)) → 𝑔 = 𝑎)
98fveq1d 6907 . . . . . . . . . . . . . 14 ((𝑔 = 𝑎𝑖 ∈ (1...𝐾)) → (𝑔𝑖) = (𝑎𝑖))
109sumeq2dv 15739 . . . . . . . . . . . . 13 (𝑔 = 𝑎 → Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = Σ𝑖 ∈ (1...𝐾)(𝑎𝑖))
1110eqeq1d 2738 . . . . . . . . . . . 12 (𝑔 = 𝑎 → (Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...𝐾)(𝑎𝑖) = 𝑁))
127, 11anbi12d 632 . . . . . . . . . . 11 (𝑔 = 𝑎 → ((𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁) ↔ (𝑎:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑎𝑖) = 𝑁)))
136, 12elab 3678 . . . . . . . . . 10 (𝑎 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)} ↔ (𝑎:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑎𝑖) = 𝑁))
145, 13sylib 218 . . . . . . . . 9 ((𝜑𝑎𝐴) → (𝑎:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑎𝑖) = 𝑁))
1514simpld 494 . . . . . . . 8 ((𝜑𝑎𝐴) → 𝑎:(1...𝐾)⟶ℕ0)
1615adantr 480 . . . . . . 7 (((𝜑𝑎𝐴) ∧ 𝑥𝑆) → 𝑎:(1...𝐾)⟶ℕ0)
17 sticksstones18.5 . . . . . . . . . . 11 (𝜑𝑍:(1...𝐾)–1-1-onto𝑆)
18 f1ocnv 6859 . . . . . . . . . . 11 (𝑍:(1...𝐾)–1-1-onto𝑆𝑍:𝑆1-1-onto→(1...𝐾))
1917, 18syl 17 . . . . . . . . . 10 (𝜑𝑍:𝑆1-1-onto→(1...𝐾))
20 f1of 6847 . . . . . . . . . 10 (𝑍:𝑆1-1-onto→(1...𝐾) → 𝑍:𝑆⟶(1...𝐾))
2119, 20syl 17 . . . . . . . . 9 (𝜑𝑍:𝑆⟶(1...𝐾))
2221adantr 480 . . . . . . . 8 ((𝜑𝑎𝐴) → 𝑍:𝑆⟶(1...𝐾))
2322ffvelcdmda 7103 . . . . . . 7 (((𝜑𝑎𝐴) ∧ 𝑥𝑆) → (𝑍𝑥) ∈ (1...𝐾))
2416, 23ffvelcdmd 7104 . . . . . 6 (((𝜑𝑎𝐴) ∧ 𝑥𝑆) → (𝑎‘(𝑍𝑥)) ∈ ℕ0)
2524fmpttd 7134 . . . . 5 ((𝜑𝑎𝐴) → (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))):𝑆⟶ℕ0)
26 eqidd 2737 . . . . . . . 8 (((𝜑𝑎𝐴) ∧ 𝑖𝑆) → (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) = (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))
27 simpr 484 . . . . . . . . . 10 ((((𝜑𝑎𝐴) ∧ 𝑖𝑆) ∧ 𝑥 = 𝑖) → 𝑥 = 𝑖)
2827fveq2d 6909 . . . . . . . . 9 ((((𝜑𝑎𝐴) ∧ 𝑖𝑆) ∧ 𝑥 = 𝑖) → (𝑍𝑥) = (𝑍𝑖))
2928fveq2d 6909 . . . . . . . 8 ((((𝜑𝑎𝐴) ∧ 𝑖𝑆) ∧ 𝑥 = 𝑖) → (𝑎‘(𝑍𝑥)) = (𝑎‘(𝑍𝑖)))
30 simpr 484 . . . . . . . 8 (((𝜑𝑎𝐴) ∧ 𝑖𝑆) → 𝑖𝑆)
31 fvexd 6920 . . . . . . . 8 (((𝜑𝑎𝐴) ∧ 𝑖𝑆) → (𝑎‘(𝑍𝑖)) ∈ V)
3226, 29, 30, 31fvmptd 7022 . . . . . . 7 (((𝜑𝑎𝐴) ∧ 𝑖𝑆) → ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖) = (𝑎‘(𝑍𝑖)))
3332sumeq2dv 15739 . . . . . 6 ((𝜑𝑎𝐴) → Σ𝑖𝑆 ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖) = Σ𝑖𝑆 (𝑎‘(𝑍𝑖)))
34 fveq2 6905 . . . . . . . . 9 (𝑛 = (𝑍𝑖) → (𝑎𝑛) = (𝑎‘(𝑍𝑖)))
35 fzfid 14015 . . . . . . . . . 10 ((𝜑𝑎𝐴) → (1...𝐾) ∈ Fin)
3617adantr 480 . . . . . . . . . . . 12 ((𝜑𝑎𝐴) → 𝑍:(1...𝐾)–1-1-onto𝑆)
37 f1oenfi 9220 . . . . . . . . . . . 12 (((1...𝐾) ∈ Fin ∧ 𝑍:(1...𝐾)–1-1-onto𝑆) → (1...𝐾) ≈ 𝑆)
3835, 36, 37syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → (1...𝐾) ≈ 𝑆)
3938ensymd 9046 . . . . . . . . . 10 ((𝜑𝑎𝐴) → 𝑆 ≈ (1...𝐾))
40 enfii 9227 . . . . . . . . . 10 (((1...𝐾) ∈ Fin ∧ 𝑆 ≈ (1...𝐾)) → 𝑆 ∈ Fin)
4135, 39, 40syl2anc 584 . . . . . . . . 9 ((𝜑𝑎𝐴) → 𝑆 ∈ Fin)
4219adantr 480 . . . . . . . . 9 ((𝜑𝑎𝐴) → 𝑍:𝑆1-1-onto→(1...𝐾))
43 eqidd 2737 . . . . . . . . 9 (((𝜑𝑎𝐴) ∧ 𝑖𝑆) → (𝑍𝑖) = (𝑍𝑖))
44 nn0sscn 12533 . . . . . . . . . . . 12 0 ⊆ ℂ
4544a1i 11 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → ℕ0 ⊆ ℂ)
46 fss 6751 . . . . . . . . . . 11 ((𝑎:(1...𝐾)⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑎:(1...𝐾)⟶ℂ)
4715, 45, 46syl2anc 584 . . . . . . . . . 10 ((𝜑𝑎𝐴) → 𝑎:(1...𝐾)⟶ℂ)
4847ffvelcdmda 7103 . . . . . . . . 9 (((𝜑𝑎𝐴) ∧ 𝑛 ∈ (1...𝐾)) → (𝑎𝑛) ∈ ℂ)
4934, 41, 42, 43, 48fsumf1o 15760 . . . . . . . 8 ((𝜑𝑎𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎𝑛) = Σ𝑖𝑆 (𝑎‘(𝑍𝑖)))
5049eqcomd 2742 . . . . . . 7 ((𝜑𝑎𝐴) → Σ𝑖𝑆 (𝑎‘(𝑍𝑖)) = Σ𝑛 ∈ (1...𝐾)(𝑎𝑛))
51 fveq2 6905 . . . . . . . . . 10 (𝑛 = 𝑖 → (𝑎𝑛) = (𝑎𝑖))
5251cbvsumv 15733 . . . . . . . . 9 Σ𝑛 ∈ (1...𝐾)(𝑎𝑛) = Σ𝑖 ∈ (1...𝐾)(𝑎𝑖)
5352a1i 11 . . . . . . . 8 ((𝜑𝑎𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎𝑛) = Σ𝑖 ∈ (1...𝐾)(𝑎𝑖))
5414simprd 495 . . . . . . . 8 ((𝜑𝑎𝐴) → Σ𝑖 ∈ (1...𝐾)(𝑎𝑖) = 𝑁)
5553, 54eqtrd 2776 . . . . . . 7 ((𝜑𝑎𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎𝑛) = 𝑁)
5650, 55eqtrd 2776 . . . . . 6 ((𝜑𝑎𝐴) → Σ𝑖𝑆 (𝑎‘(𝑍𝑖)) = 𝑁)
5733, 56eqtrd 2776 . . . . 5 ((𝜑𝑎𝐴) → Σ𝑖𝑆 ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖) = 𝑁)
5825, 57jca 511 . . . 4 ((𝜑𝑎𝐴) → ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))):𝑆⟶ℕ0 ∧ Σ𝑖𝑆 ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖) = 𝑁))
59 fzfid 14015 . . . . . . . 8 (𝜑 → (1...𝐾) ∈ Fin)
6059adantr 480 . . . . . . 7 ((𝜑𝑎𝐴) → (1...𝐾) ∈ Fin)
6159, 17, 37syl2anc 584 . . . . . . . . 9 (𝜑 → (1...𝐾) ≈ 𝑆)
6261ensymd 9046 . . . . . . . 8 (𝜑𝑆 ≈ (1...𝐾))
6362adantr 480 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑆 ≈ (1...𝐾))
6460, 63, 40syl2anc 584 . . . . . 6 ((𝜑𝑎𝐴) → 𝑆 ∈ Fin)
6564mptexd 7245 . . . . 5 ((𝜑𝑎𝐴) → (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) ∈ V)
66 feq1 6715 . . . . . . 7 ( = (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) → (:𝑆⟶ℕ0 ↔ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))):𝑆⟶ℕ0))
67 simpl 482 . . . . . . . . . 10 (( = (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) ∧ 𝑖𝑆) → = (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))
6867fveq1d 6907 . . . . . . . . 9 (( = (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) ∧ 𝑖𝑆) → (𝑖) = ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖))
6968sumeq2dv 15739 . . . . . . . 8 ( = (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) → Σ𝑖𝑆 (𝑖) = Σ𝑖𝑆 ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖))
7069eqeq1d 2738 . . . . . . 7 ( = (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) → (Σ𝑖𝑆 (𝑖) = 𝑁 ↔ Σ𝑖𝑆 ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖) = 𝑁))
7166, 70anbi12d 632 . . . . . 6 ( = (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) → ((:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁) ↔ ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))):𝑆⟶ℕ0 ∧ Σ𝑖𝑆 ((𝑥𝑆 ↦ (𝑎‘(𝑍𝑥)))‘𝑖) = 𝑁)))
7271elabg 3675 . . . . 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 257 . . 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 2843 . 2 ((𝜑𝑎𝐴) → (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))) ∈ 𝐵)
78 sticksstones18.6 . 2 𝐹 = (𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))
7977, 78fmptd 7133 1 (𝜑𝐹:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  {cab 2713  Vcvv 3479  wss 3950   class class class wbr 5142  cmpt 5224  ccnv 5683  wf 6556  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432  cen 8983  Fincfn 8986  cc 11154  1c1 11157  0cn0 12528  ...cfz 13548  Σcsu 15723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-rp 13036  df-fz 13549  df-fzo 13696  df-seq 14044  df-exp 14104  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-clim 15525  df-sum 15724
This theorem is referenced by:  sticksstones19  42167
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