Theorem ballotlemrinv0 34500

Description: Lemma for ballotlemrinv 34501. (Contributed by Thierry Arnoux, 18-Apr-2017.)

Hypotheses

Ref	Expression
ballotth.m	⊢ 𝑀 ∈ ℕ
ballotth.n	⊢ 𝑁 ∈ ℕ
ballotth.o	⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
ballotth.p	⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
ballotth.f	⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotth.e	⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
ballotth.mgtn	⊢ 𝑁 < 𝑀
ballotth.i	⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
ballotth.s	⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
ballotth.r	⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))

Assertion

Ref	Expression
ballotlemrinv0	⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (𝐷 ∈ (𝑂 ∖ 𝐸) ∧ 𝐶 = ((𝑆‘𝐷) “ 𝐷)))

Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐   𝑖,𝑀   𝑖,𝑁   𝑖,𝑂   𝑘,𝑀   𝑘,𝑁   𝑘,𝑂   𝑖,𝑐,𝐹,𝑘   𝐶,𝑖,𝑘   𝑖,𝐸,𝑘   𝐶,𝑘   𝑘,𝐼,𝑐   𝐸,𝑐   𝑖,𝐼,𝑐   𝑆,𝑘   𝐷,𝑖,𝑘   𝑆,𝑖,𝑐   𝑅,𝑖,𝑘   𝑥,𝑐   𝑥,𝐶   𝑥,𝐹   𝑥,𝑀   𝑥,𝑁,𝑖,𝑘

Allowed substitution hints:   𝐶(𝑐)   𝐷(𝑥,𝑐)   𝑃(𝑥,𝑖,𝑘,𝑐)   𝑅(𝑥,𝑐)   𝑆(𝑥)   𝐸(𝑥)   𝐼(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemrinv0

Step	Hyp	Ref	Expression
1		ballotth.m	. . . . . 6 ⊢ 𝑀 ∈ ℕ
2		ballotth.n	. . . . . 6 ⊢ 𝑁 ∈ ℕ
3		ballotth.o	. . . . . 6 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
4		ballotth.p	. . . . . 6 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
5		ballotth.f	. . . . . 6 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6		ballotth.e	. . . . . 6 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
7		ballotth.mgtn	. . . . . 6 ⊢ 𝑁 < 𝑀
8		ballotth.i	. . . . . 6 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
9		ballotth.s	. . . . . 6 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
10		ballotth.r	. . . . . 6 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotlemrval 34485	. . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) = ((𝑆‘𝐶) “ 𝐶))
12	11	adantr 480	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (𝑅‘𝐶) = ((𝑆‘𝐶) “ 𝐶))
13		simpr 484	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → 𝐷 = ((𝑆‘𝐶) “ 𝐶))
14	12, 13	eqtr4d 2767	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (𝑅‘𝐶) = 𝐷)
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotlemrc 34498	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ (𝑂 ∖ 𝐸))
16	15	adantr 480	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (𝑅‘𝐶) ∈ (𝑂 ∖ 𝐸))
17	14, 16	eqeltrrd 2829	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → 𝐷 ∈ (𝑂 ∖ 𝐸))
18	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsf1o 34481	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ ◡(𝑆‘𝐶) = (𝑆‘𝐶)))
19	18	simprd 495	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ◡(𝑆‘𝐶) = (𝑆‘𝐶))
20	19	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → ◡(𝑆‘𝐶) = (𝑆‘𝐶))
21	20	eqcomd 2735	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (𝑆‘𝐶) = ◡(𝑆‘𝐶))
22	21, 13	imaeq12d 6016	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → ((𝑆‘𝐶) “ 𝐷) = (◡(𝑆‘𝐶) “ ((𝑆‘𝐶) “ 𝐶)))
23		simpl 482	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → 𝐶 ∈ (𝑂 ∖ 𝐸))
24	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotlemirc 34499	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘(𝑅‘𝐶)) = (𝐼‘𝐶))
25	24	adantr 480	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (𝐼‘(𝑅‘𝐶)) = (𝐼‘𝐶))
26	14	fveq2d 6830	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (𝐼‘(𝑅‘𝐶)) = (𝐼‘𝐷))
27	25, 26	eqtr3d 2766	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (𝐼‘𝐶) = (𝐼‘𝐷))
28	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemieq 34484	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) = (𝐼‘𝐷)) → (𝑆‘𝐶) = (𝑆‘𝐷))
29	23, 17, 27, 28	syl3anc 1373	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (𝑆‘𝐶) = (𝑆‘𝐷))
30	29	imaeq1d 6014	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → ((𝑆‘𝐶) “ 𝐷) = ((𝑆‘𝐷) “ 𝐷))
31	18	simpld 494	. . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
32		f1of1 6767	. . . . 5 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
33	23, 31, 32	3syl 18	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
34		eldifi 4084	. . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
35	1, 2, 3	ballotlemelo 34455	. . . . . 6 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
36	35	simplbi 497	. . . . 5 ⊢ (𝐶 ∈ 𝑂 → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
37	23, 34, 36	3syl 18	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
38		f1imacnv 6784	. . . 4 ⊢ (((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)) ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁))) → (◡(𝑆‘𝐶) “ ((𝑆‘𝐶) “ 𝐶)) = 𝐶)
39	33, 37, 38	syl2anc 584	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (◡(𝑆‘𝐶) “ ((𝑆‘𝐶) “ 𝐶)) = 𝐶)
40	22, 30, 39	3eqtr3rd 2773	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → 𝐶 = ((𝑆‘𝐷) “ 𝐷))
41	17, 40	jca 511	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (𝐷 ∈ (𝑂 ∖ 𝐸) ∧ 𝐶 = ((𝑆‘𝐷) “ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3396   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  ifcif 4478  𝒫 cpw 4553   class class class wbr 5095   ↦ cmpt 5176  ^◡ccnv 5622   “ cima 5626  –1-1→wf1 6483  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353  infcinf 9350  ℝcr 11027  0cc0 11028  1c1 11029   + caddc 11031   < clt 11168   ≤ cle 11169   − cmin 11365   / cdiv 11795  ℕcn 12146  ℤcz 12489  ...cfz 13428  ♯chash 14255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9351  df-inf 9352  df-dju 9816  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-n0 12403  df-z 12490  df-uz 12754  df-rp 12912  df-fz 13429  df-hash 14256

This theorem is referenced by:  ballotlemrinv  34501