Theorem metakunt11 40587

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 2740	. . . . 5 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 = 𝐼 ↔ (𝐶‘𝑋) = 𝐼))
4		breq1 5108	. . . . . 6 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 < 𝐼 ↔ (𝐶‘𝑋) < 𝐼))
5		id 22	. . . . . 6 ⊢ (𝑥 = (𝐶‘𝑋) → 𝑥 = (𝐶‘𝑋))
6		oveq1 7364	. . . . . 6 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 − 1) = ((𝐶‘𝑋) − 1))
7	4, 5, 6	ifbieq12d 4514	. . . . 5 ⊢ (𝑥 = (𝐶‘𝑋) → if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) = if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)))
8	3, 7	ifbieq2d 4512	. . . 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 482	. . 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 2740	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (𝑦 = 𝑀 ↔ 𝑋 = 𝑀))
13		breq1 5108	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → (𝑦 < 𝐼 ↔ 𝑋 < 𝐼))
14		id 22	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋)
15		oveq1 7364	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → (𝑦 + 1) = (𝑋 + 1))
16	13, 14, 15	ifbieq12d 4514	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)) = if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)))
17	12, 16	ifbieq2d 4512	. . . . . . . . 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 482	. . . . . . . 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 13470	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ (1...𝑀) → 𝑋 ∈ ℕ)
21	19, 20	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ ℕ)
22	21	nnred 12168	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℝ)
23	22	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 ∈ ℝ)
24		metakunt11.2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ ℕ)
25	24	nnred 12168	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ∈ ℝ)
26	25	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝐼 ∈ ℝ)
27		metakunt11.1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℕ)
28	27	nnred 12168	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℝ)
29	28	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑀 ∈ ℝ)
30		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 < 𝐼)
31		metakunt11.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ≤ 𝑀)
32	31	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝐼 ≤ 𝑀)
33	23, 26, 29, 30, 32	ltletrd 11315	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 < 𝑀)
34	23, 33	ltned 11291	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 ≠ 𝑀)
35		df-ne 2944	. . . . . . . . . . . 12 ⊢ (𝑋 ≠ 𝑀 ↔ ¬ 𝑋 = 𝑀)
36	34, 35	sylib 217	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → ¬ 𝑋 = 𝑀)
37		iffalse 4495	. . . . . . . . . . 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 4492	. . . . . . . . . . 11 ⊢ (𝑋 < 𝐼 → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = 𝑋)
40	39	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = 𝑋)
41	38, 40	eqtrd 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = 𝑋)
42	41	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 < 𝐼) ∧ 𝑦 = 𝑋) → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = 𝑋)
43	18, 42	eqtrd 2776	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 < 𝐼) ∧ 𝑦 = 𝑋) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝑋)
44	19	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 ∈ (1...𝑀))
45	11, 43, 44, 44	fvmptd 6955	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → (𝐶‘𝑋) = 𝑋)
46		eqeq1 2740	. . . . . . 7 ⊢ ((𝐶‘𝑋) = 𝑋 → ((𝐶‘𝑋) = 𝐼 ↔ 𝑋 = 𝐼))
47	46	ifbid 4509	. . . . . 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 11291	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 ≠ 𝐼)
50	49	neneqd 2948	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → ¬ 𝑋 = 𝐼)
51		iffalse 4495	. . . . . . 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 2742	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 = (𝐶‘𝑋))
54		breq1 5108	. . . . . . . . . 10 ⊢ (𝑋 = (𝐶‘𝑋) → (𝑋 < 𝐼 ↔ (𝐶‘𝑋) < 𝐼))
55		id 22	. . . . . . . . . 10 ⊢ (𝑋 = (𝐶‘𝑋) → 𝑋 = (𝐶‘𝑋))
56		oveq1 7364	. . . . . . . . . 10 ⊢ (𝑋 = (𝐶‘𝑋) → (𝑋 − 1) = ((𝐶‘𝑋) − 1))
57	54, 55, 56	ifbieq12d 4514	. . . . . . . . 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 2742	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)) = if(𝑋 < 𝐼, 𝑋, (𝑋 − 1)))
60	30	iftrued 4494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 < 𝐼, 𝑋, (𝑋 − 1)) = 𝑋)
61	59, 60	eqtrd 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)) = 𝑋)
62	52, 61	eqtrd 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑋)
63	48, 62	eqtrd 2776	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑋)
64	63	adantr 481	. . 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 2776	. 2 ⊢ (((𝜑 ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝐶‘𝑋)) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑋)
66	27, 24, 31, 10	metakunt2 40578	. . . 4 ⊢ (𝜑 → 𝐶:(1...𝑀)⟶(1...𝑀))
67	66	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝐶:(1...𝑀)⟶(1...𝑀))
68	67, 44	ffvelcdmd 7036	. 2 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → (𝐶‘𝑋) ∈ (1...𝑀))
69	2, 65, 68, 44	fvmptd 6955	1 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → (𝐴‘(𝐶‘𝑋)) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ifcif 4486   class class class wbr 5105   ↦ cmpt 5188  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  ℝcr 11050  1c1 11052   + caddc 11054   < clt 11189   ≤ cle 11190   − cmin 11385  ℕcn 12153  ...cfz 13424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425

This theorem is referenced by:  metakunt13  40589