Theorem metakunt11 40063

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 2742	. . . . 5 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 = 𝐼 ↔ (𝐶‘𝑋) = 𝐼))
4		breq1 5073	. . . . . 6 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 < 𝐼 ↔ (𝐶‘𝑋) < 𝐼))
5		id 22	. . . . . 6 ⊢ (𝑥 = (𝐶‘𝑋) → 𝑥 = (𝐶‘𝑋))
6		oveq1 7262	. . . . . 6 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 − 1) = ((𝐶‘𝑋) − 1))
7	4, 5, 6	ifbieq12d 4484	. . . . 5 ⊢ (𝑥 = (𝐶‘𝑋) → if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) = if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)))
8	3, 7	ifbieq2d 4482	. . . 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 2742	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (𝑦 = 𝑀 ↔ 𝑋 = 𝑀))
13		breq1 5073	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → (𝑦 < 𝐼 ↔ 𝑋 < 𝐼))
14		id 22	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋)
15		oveq1 7262	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → (𝑦 + 1) = (𝑋 + 1))
16	13, 14, 15	ifbieq12d 4484	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)) = if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)))
17	12, 16	ifbieq2d 4482	. . . . . . . . 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 13214	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ (1...𝑀) → 𝑋 ∈ ℕ)
21	19, 20	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ ℕ)
22	21	nnred 11918	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℝ)
23	22	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 ∈ ℝ)
24		metakunt11.2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ ℕ)
25	24	nnred 11918	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ∈ ℝ)
26	25	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝐼 ∈ ℝ)
27		metakunt11.1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℕ)
28	27	nnred 11918	. . . . . . . . . . . . . . 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 11065	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 < 𝑀)
34	23, 33	ltned 11041	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 ≠ 𝑀)
35		df-ne 2943	. . . . . . . . . . . 12 ⊢ (𝑋 ≠ 𝑀 ↔ ¬ 𝑋 = 𝑀)
36	34, 35	sylib 217	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → ¬ 𝑋 = 𝑀)
37		iffalse 4465	. . . . . . . . . . 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 4462	. . . . . . . . . . 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 2778	. . . . . . . . 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 2778	. . . . . . 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 6864	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → (𝐶‘𝑋) = 𝑋)
46		eqeq1 2742	. . . . . . 7 ⊢ ((𝐶‘𝑋) = 𝑋 → ((𝐶‘𝑋) = 𝐼 ↔ 𝑋 = 𝐼))
47	46	ifbid 4479	. . . . . 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 11041	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 ≠ 𝐼)
50	49	neneqd 2947	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → ¬ 𝑋 = 𝐼)
51		iffalse 4465	. . . . . . 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 2744	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝑋 = (𝐶‘𝑋))
54		breq1 5073	. . . . . . . . . 10 ⊢ (𝑋 = (𝐶‘𝑋) → (𝑋 < 𝐼 ↔ (𝐶‘𝑋) < 𝐼))
55		id 22	. . . . . . . . . 10 ⊢ (𝑋 = (𝐶‘𝑋) → 𝑋 = (𝐶‘𝑋))
56		oveq1 7262	. . . . . . . . . 10 ⊢ (𝑋 = (𝐶‘𝑋) → (𝑋 − 1) = ((𝐶‘𝑋) − 1))
57	54, 55, 56	ifbieq12d 4484	. . . . . . . . 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 2744	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)) = if(𝑋 < 𝐼, 𝑋, (𝑋 − 1)))
60	30	iftrued 4464	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 < 𝐼, 𝑋, (𝑋 − 1)) = 𝑋)
61	59, 60	eqtrd 2778	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)) = 𝑋)
62	52, 61	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → if(𝑋 = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑋)
63	48, 62	eqtrd 2778	. . . 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 2778	. 2 ⊢ (((𝜑 ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝐶‘𝑋)) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑋)
66	27, 24, 31, 10	metakunt2 40054	. . . 4 ⊢ (𝜑 → 𝐶:(1...𝑀)⟶(1...𝑀))
67	66	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → 𝐶:(1...𝑀)⟶(1...𝑀))
68	67, 44	ffvelrnd 6944	. 2 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → (𝐶‘𝑋) ∈ (1...𝑀))
69	2, 65, 68, 44	fvmptd 6864	1 ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → (𝐴‘(𝐶‘𝑋)) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ifcif 4456   class class class wbr 5070   ↦ cmpt 5153  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  ℝcr 10801  1c1 10803   + caddc 10805   < clt 10940   ≤ cle 10941   − cmin 11135  ℕcn 11903  ...cfz 13168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169

This theorem is referenced by:  metakunt13  40065