Theorem ballotlemrinv0 34536

Description: Lemma for ballotlemrinv 34537. (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 34521	. . . . 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 2768	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (𝑅‘𝐶) = 𝐷)
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotlemrc 34534	. . . 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 2830	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → 𝐷 ∈ (𝑂 ∖ 𝐸))
18	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsf1o 34517	. . . . . . 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 2736	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (𝑆‘𝐶) = ◡(𝑆‘𝐶))
22	21, 13	imaeq12d 6007	. . 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 34535	. . . . . . 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 6821	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (𝐼‘(𝑅‘𝐶)) = (𝐼‘𝐷))
27	25, 26	eqtr3d 2767	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (𝐼‘𝐶) = (𝐼‘𝐷))
28	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemieq 34520	. . . . 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 6005	. . 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 6758	. . . . 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 4079	. . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
35	1, 2, 3	ballotlemelo 34491	. . . . . 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 6775	. . . 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 2774	. 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 1541   ∈ wcel 2110  ∀wral 3045  {crab 3393   ∖ cdif 3897   ∩ cin 3899   ⊆ wss 3900  ifcif 4473  𝒫 cpw 4548   class class class wbr 5089   ↦ cmpt 5170  ^◡ccnv 5613   “ cima 5617  –1-1→wf1 6474  –1-1-onto→wf1o 6476  ‘cfv 6477  (class class class)co 7341  infcinf 9320  ℝcr 10997  0cc0 10998  1c1 10999   + caddc 11001   < clt 11138   ≤ cle 11139   − cmin 11336   / cdiv 11766  ℕcn 12117  ℤcz 12460  ...cfz 13399  ♯chash 14229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-oadd 8384  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-sup 9321  df-inf 9322  df-dju 9786  df-card 9824  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-n0 12374  df-z 12461  df-uz 12725  df-rp 12883  df-fz 13400  df-hash 14230

This theorem is referenced by:  ballotlemrinv  34537