Theorem metakunt11 39642

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 2763	. . . . 5 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 = 𝐼 ↔ (𝐶‘𝑋) = 𝐼))
4		breq1 5028	. . . . . 6 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 < 𝐼 ↔ (𝐶‘𝑋) < 𝐼))
5		id 22	. . . . . 6 ⊢ (𝑥 = (𝐶‘𝑋) → 𝑥 = (𝐶‘𝑋))
6		oveq1 7150	. . . . . 6 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 − 1) = ((𝐶‘𝑋) − 1))
7	4, 5, 6	ifbieq12d 4441	. . . . 5 ⊢ (𝑥 = (𝐶‘𝑋) → if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) = if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)))
8	3, 7	ifbieq2d 4439	. . . 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 486	. . 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 2763	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (𝑦 = 𝑀 ↔ 𝑋 = 𝑀))
13		breq1 5028	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → (𝑦 < 𝐼 ↔ 𝑋 < 𝐼))
14		id 22	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋)
15		oveq1 7150	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → (𝑦 + 1) = (𝑋 + 1))
16	13, 14, 15	ifbieq12d 4441	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)) = if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)))
17	12, 16	ifbieq2d 4439	. . . . . . . . 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 486	. . . . . . . 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 12970	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ (1...𝑀) → 𝑋 ∈ ℕ)
21	19, 20	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ ℕ)
22	21	nnred 11674	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℝ)
23	22	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 ∈ ℝ)
24		metakunt11.2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ ℕ)
25	24	nnred 11674	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ∈ ℝ)
26	25	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝐼 ∈ ℝ)
27		metakunt11.1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℕ)
28	27	nnred 11674	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℝ)
29	28	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑀 ∈ ℝ)
30		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 < 𝐼)
31		metakunt11.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ≤ 𝑀)
32	31	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝐼 ≤ 𝑀)
33	23, 26, 29, 30, 32	ltletrd 10823	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 < 𝑀)
34	23, 33	ltned 10799	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 ≠ 𝑀)
35		df-ne 2950	. . . . . . . . . . . 12 ⊢ (𝑋 ≠ 𝑀 ↔ ¬ 𝑋 = 𝑀)
36	34, 35	sylib 221	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → ¬ 𝑋 = 𝑀)
37		iffalse 4422	. . . . . . . . . . 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 4419	. . . . . . . . . . 11 ⊢ (𝑋 < 𝐼 → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = 𝑋)
40	39	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = 𝑋)
41	38, 40	eqtrd 2794	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = 𝑋)
42	41	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 < 𝐼) ∧ 𝑦 = 𝑋) → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = 𝑋)
43	18, 42	eqtrd 2794	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 < 𝐼) ∧ 𝑦 = 𝑋) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝑋)
44	19	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 ∈ (1...𝑀))
45	11, 43, 44, 44	fvmptd 6759	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → (𝐶‘𝑋) = 𝑋)
46		eqeq1 2763	. . . . . . 7 ⊢ ((𝐶‘𝑋) = 𝑋 → ((𝐶‘𝑋) = 𝐼 ↔ 𝑋 = 𝐼))
47	46	ifbid 4436	. . . . . 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 10799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 ≠ 𝐼)
50	49	neneqd 2954	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → ¬ 𝑋 = 𝐼)
51		iffalse 4422	. . . . . . 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 2765	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 = (𝐶‘𝑋))
54		breq1 5028	. . . . . . . . . 10 ⊢ (𝑋 = (𝐶‘𝑋) → (𝑋 < 𝐼 ↔ (𝐶‘𝑋) < 𝐼))
55		id 22	. . . . . . . . . 10 ⊢ (𝑋 = (𝐶‘𝑋) → 𝑋 = (𝐶‘𝑋))
56		oveq1 7150	. . . . . . . . . 10 ⊢ (𝑋 = (𝐶‘𝑋) → (𝑋 − 1) = ((𝐶‘𝑋) − 1))
57	54, 55, 56	ifbieq12d 4441	. . . . . . . . 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 2765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)) = if(𝑋 < 𝐼, 𝑋, (𝑋 − 1)))
60	30	iftrued 4421	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 < 𝐼, 𝑋, (𝑋 − 1)) = 𝑋)
61	59, 60	eqtrd 2794	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)) = 𝑋)
62	52, 61	eqtrd 2794	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑋)
63	48, 62	eqtrd 2794	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑋)
64	63	adantr 485	. . 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 2794	. 2 ⊢ (((𝜑 ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝐶‘𝑋)) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑋)
66	27, 24, 31, 10	metakunt2 39633	. . . 4 ⊢ (𝜑 → 𝐶:(1...𝑀)⟶(1...𝑀))
67	66	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝐶:(1...𝑀)⟶(1...𝑀))
68	67, 44	ffvelrnd 6836	. 2 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → (𝐶‘𝑋) ∈ (1...𝑀))
69	2, 65, 68, 44	fvmptd 6759	1 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → (𝐴‘(𝐶‘𝑋)) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112   ≠ wne 2949  ifcif 4413   class class class wbr 5025   ↦ cmpt 5105  ⟶wf 6324  ‘cfv 6328  (class class class)co 7143  ℝcr 10559  1c1 10561   + caddc 10563   < clt 10698   ≤ cle 10699   − cmin 10893  ℕcn 11659  ...cfz 12924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452  ax-cnex 10616  ax-resscn 10617  ax-1cn 10618  ax-icn 10619  ax-addcl 10620  ax-addrcl 10621  ax-mulcl 10622  ax-mulrcl 10623  ax-mulcom 10624  ax-addass 10625  ax-mulass 10626  ax-distr 10627  ax-i2m1 10628  ax-1ne0 10629  ax-1rid 10630  ax-rnegex 10631  ax-rrecex 10632  ax-cnre 10633  ax-pre-lttri 10634  ax-pre-lttrn 10635  ax-pre-ltadd 10636  ax-pre-mulgt0 10637

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-nel 3054  df-ral 3073  df-rex 3074  df-reu 3075  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-tp 4520  df-op 4522  df-uni 4792  df-iun 4878  df-br 5026  df-opab 5088  df-mpt 5106  df-tr 5132  df-id 5423  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-we 5478  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7101  df-ov 7146  df-oprab 7147  df-mpo 7148  df-om 7573  df-1st 7686  df-2nd 7687  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-er 8292  df-en 8521  df-dom 8522  df-sdom 8523  df-pnf 10700  df-mnf 10701  df-xr 10702  df-ltxr 10703  df-le 10704  df-sub 10895  df-neg 10896  df-nn 11660  df-n0 11920  df-z 12006  df-uz 12268  df-fz 12925

This theorem is referenced by:  metakunt13  39644