Theorem metakunt11 42228

Description: C is the right inverse for A. (Contributed by metakunt, 24-May-2024.)

Hypotheses

Ref	Expression
metakunt11.1	⊢ (𝜑 → 𝑀 ∈ ℕ)
metakunt11.2	⊢ (𝜑 → 𝐼 ∈ ℕ)
metakunt11.3	⊢ (𝜑 → 𝐼 ≤ 𝑀)
metakunt11.4	⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))
metakunt11.5	⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))
metakunt11.6	⊢ (𝜑 → 𝑋 ∈ (1...𝑀))

Assertion

Ref	Expression
metakunt11	⊢ ((𝜑 ∧ 𝑋 < 𝐼) → (𝐴‘(𝐶‘𝑋)) = 𝑋)

Distinct variable groups:   𝑥,𝐶   𝑥,𝐼   𝑦,𝐼   𝑥,𝑀   𝑦,𝑀   𝑥,𝑋   𝑦,𝑋   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐶(𝑦)

Proof of Theorem metakunt11

Step	Hyp	Ref	Expression
1		metakunt11.4	. . 3 ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))
2	1	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))))
3		eqeq1 2739	. . . . 5 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 = 𝐼 ↔ (𝐶‘𝑋) = 𝐼))
4		breq1 5122	. . . . . 6 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 < 𝐼 ↔ (𝐶‘𝑋) < 𝐼))
5		id 22	. . . . . 6 ⊢ (𝑥 = (𝐶‘𝑋) → 𝑥 = (𝐶‘𝑋))
6		oveq1 7412	. . . . . 6 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 − 1) = ((𝐶‘𝑋) − 1))
7	4, 5, 6	ifbieq12d 4529	. . . . 5 ⊢ (𝑥 = (𝐶‘𝑋) → if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) = if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)))
8	3, 7	ifbieq2d 4527	. . . 4 ⊢ (𝑥 = (𝐶‘𝑋) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))))
9	8	adantl 481	. . 3 ⊢ (((𝜑 ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝐶‘𝑋)) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))))
10		metakunt11.5	. . . . . . . 8 ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))
11	10	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))))
12		eqeq1 2739	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (𝑦 = 𝑀 ↔ 𝑋 = 𝑀))
13		breq1 5122	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → (𝑦 < 𝐼 ↔ 𝑋 < 𝐼))
14		id 22	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋)
15		oveq1 7412	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → (𝑦 + 1) = (𝑋 + 1))
16	13, 14, 15	ifbieq12d 4529	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)) = if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)))
17	12, 16	ifbieq2d 4527	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))))
18	17	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 < 𝐼) ∧ 𝑦 = 𝑋) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))))
19		metakunt11.6	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ∈ (1...𝑀))
20		elfznn 13570	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ (1...𝑀) → 𝑋 ∈ ℕ)
21	19, 20	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ ℕ)
22	21	nnred 12255	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℝ)
23	22	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 ∈ ℝ)
24		metakunt11.2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ ℕ)
25	24	nnred 12255	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ∈ ℝ)
26	25	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝐼 ∈ ℝ)
27		metakunt11.1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℕ)
28	27	nnred 12255	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℝ)
29	28	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑀 ∈ ℝ)
30		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 < 𝐼)
31		metakunt11.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ≤ 𝑀)
32	31	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝐼 ≤ 𝑀)
33	23, 26, 29, 30, 32	ltletrd 11395	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 < 𝑀)
34	23, 33	ltned 11371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 ≠ 𝑀)
35		df-ne 2933	. . . . . . . . . . . 12 ⊢ (𝑋 ≠ 𝑀 ↔ ¬ 𝑋 = 𝑀)
36	34, 35	sylib 218	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → ¬ 𝑋 = 𝑀)
37		iffalse 4509	. . . . . . . . . . 11 ⊢ (¬ 𝑋 = 𝑀 → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)))
38	36, 37	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)))
39		iftrue 4506	. . . . . . . . . . 11 ⊢ (𝑋 < 𝐼 → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = 𝑋)
40	39	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = 𝑋)
41	38, 40	eqtrd 2770	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = 𝑋)
42	41	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 < 𝐼) ∧ 𝑦 = 𝑋) → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = 𝑋)
43	18, 42	eqtrd 2770	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 < 𝐼) ∧ 𝑦 = 𝑋) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝑋)
44	19	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 ∈ (1...𝑀))
45	11, 43, 44, 44	fvmptd 6993	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → (𝐶‘𝑋) = 𝑋)
46		eqeq1 2739	. . . . . . 7 ⊢ ((𝐶‘𝑋) = 𝑋 → ((𝐶‘𝑋) = 𝐼 ↔ 𝑋 = 𝐼))
47	46	ifbid 4524	. . . . . 6 ⊢ ((𝐶‘𝑋) = 𝑋 → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = if(𝑋 = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))))
48	45, 47	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = if(𝑋 = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))))
49	23, 30	ltned 11371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 ≠ 𝐼)
50	49	neneqd 2937	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → ¬ 𝑋 = 𝐼)
51		iffalse 4509	. . . . . . 7 ⊢ (¬ 𝑋 = 𝐼 → if(𝑋 = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)))
52	50, 51	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)))
53	45	eqcomd 2741	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 = (𝐶‘𝑋))
54		breq1 5122	. . . . . . . . . 10 ⊢ (𝑋 = (𝐶‘𝑋) → (𝑋 < 𝐼 ↔ (𝐶‘𝑋) < 𝐼))
55		id 22	. . . . . . . . . 10 ⊢ (𝑋 = (𝐶‘𝑋) → 𝑋 = (𝐶‘𝑋))
56		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑋 = (𝐶‘𝑋) → (𝑋 − 1) = ((𝐶‘𝑋) − 1))
57	54, 55, 56	ifbieq12d 4529	. . . . . . . . 9 ⊢ (𝑋 = (𝐶‘𝑋) → if(𝑋 < 𝐼, 𝑋, (𝑋 − 1)) = if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)))
58	53, 57	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 < 𝐼, 𝑋, (𝑋 − 1)) = if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)))
59	58	eqcomd 2741	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)) = if(𝑋 < 𝐼, 𝑋, (𝑋 − 1)))
60	30	iftrued 4508	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 < 𝐼, 𝑋, (𝑋 − 1)) = 𝑋)
61	59, 60	eqtrd 2770	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)) = 𝑋)
62	52, 61	eqtrd 2770	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑋)
63	48, 62	eqtrd 2770	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑋)
64	63	adantr 480	. . 3 ⊢ (((𝜑 ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝐶‘𝑋)) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑋)
65	9, 64	eqtrd 2770	. 2 ⊢ (((𝜑 ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝐶‘𝑋)) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑋)
66	27, 24, 31, 10	metakunt2 42219	. . . 4 ⊢ (𝜑 → 𝐶:(1...𝑀)⟶(1...𝑀))
67	66	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝐶:(1...𝑀)⟶(1...𝑀))
68	67, 44	ffvelcdmd 7075	. 2 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → (𝐶‘𝑋) ∈ (1...𝑀))
69	2, 65, 68, 44	fvmptd 6993	1 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → (𝐴‘(𝐶‘𝑋)) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ifcif 4500   class class class wbr 5119   ↦ cmpt 5201  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  ℝcr 11128  1c1 11130   + caddc 11132   < clt 11269   ≤ cle 11270   − cmin 11466  ℕcn 12240  ...cfz 13524

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-n0 12502  df-z 12589  df-uz 12853  df-fz 13525

This theorem is referenced by:  metakunt13  42230