metakunt31 - Mathbox for metakunt

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Theorem metakunt31 39662

Description: Construction of one solution of the increment equation system. (Contributed by metakunt, 18-Jul-2024.)

**Hypotheses**
Ref	Expression
metakunt31.1	⊢ (𝜑 → 𝑀 ∈ ℕ)
metakunt31.2	⊢ (𝜑 → 𝐼 ∈ ℕ)
metakunt31.3	⊢ (𝜑 → 𝐼 ≤ 𝑀)
metakunt31.4	⊢ (𝜑 → 𝑋 ∈ (1...𝑀))
metakunt31.5	⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))
metakunt31.6	⊢ 𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼)))))
metakunt31.7	⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))
metakunt31.8	⊢ 𝐺 = if(𝐼 ≤ (𝑋 + (𝑀 − 𝐼)), 1, 0)
metakunt31.9	⊢ 𝐻 = if(𝐼 ≤ (𝑋 − 𝐼), 1, 0)
metakunt31.10	⊢ 𝑅 = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻)))

**Assertion**
Ref	Expression
metakunt31	⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅)

Distinct variable groups: 𝑦,𝐺 𝑦,𝐻 𝑥,𝐼 𝑦,𝐼 𝑧,𝐼 𝑥,𝑀 𝑦,𝑀 𝑧,𝑀 𝑥,𝑋 𝑦,𝑋 𝑧,𝑋 𝜑,𝑥 𝜑,𝑦 𝜑,𝑧 Allowed substitution hints: 𝐴(𝑥,𝑦,𝑧) 𝐵(𝑥,𝑦,𝑧) 𝐶(𝑥,𝑦,𝑧) 𝑅(𝑥,𝑦,𝑧) 𝐺(𝑥,𝑧) 𝐻(𝑥,𝑧)

**Proof of Theorem metakunt31**
Step	Hyp	Ref	Expression
1		metakunt31.1	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
2	1	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → 𝑀 ∈ ℕ)
3		metakunt31.2	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℕ)
4	3	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → 𝐼 ∈ ℕ)
5		metakunt31.3	. . . . . 6 ⊢ (𝜑 → 𝐼 ≤ 𝑀)
6	5	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → 𝐼 ≤ 𝑀)
7		metakunt31.5	. . . . 5 ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))
8		metakunt31.7	. . . . 5 ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))
9		metakunt31.6	. . . . 5 ⊢ 𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼)))))
10		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → 𝑋 = 𝐼)
11	2, 4, 6, 7, 8, 9, 10	metakunt26 39657	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑋)
12	10	iftrued 4421	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))) = 𝑋)
13	12	eqcomd 2765	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → 𝑋 = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))))
14	11, 13	eqtrd 2794	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))))
15		metakunt31.10	. . . . 5 ⊢ 𝑅 = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻)))
16	15	eqcomi 2768	. . . 4 ⊢ if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))) = 𝑅
17	16	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))) = 𝑅)
18	14, 17	eqtrd 2794	. 2 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅)
19	1	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → 𝑀 ∈ ℕ)
20	3	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → 𝐼 ∈ ℕ)
21	5	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → 𝐼 ≤ 𝑀)
22		metakunt31.4	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (1...𝑀))
23	22	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → 𝑋 ∈ (1...𝑀))
24		simp2 1135	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → ¬ 𝑋 = 𝐼)
25		simp3 1136	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → 𝑋 < 𝐼)
26		metakunt31.8	. . . . . . 7 ⊢ 𝐺 = if(𝐼 ≤ (𝑋 + (𝑀 − 𝐼)), 1, 0)
27	19, 20, 21, 23, 7, 9, 24, 25, 8, 26	metakunt29 39660	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = ((𝑋 + (𝑀 − 𝐼)) + 𝐺))
28	24	iffalsed 4424	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))) = if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻)))
29	25	iftrued 4421	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻)) = ((𝑋 + (𝑀 − 𝐼)) + 𝐺))
30	28, 29	eqtrd 2794	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))) = ((𝑋 + (𝑀 − 𝐼)) + 𝐺))
31	30	eqcomd 2765	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → ((𝑋 + (𝑀 − 𝐼)) + 𝐺) = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))))
32	27, 31	eqtrd 2794	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))))
33	16	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))) = 𝑅)
34	32, 33	eqtrd 2794	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅)
35	34	3expa 1116	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝐼) ∧ 𝑋 < 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅)
36	1	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → 𝑀 ∈ ℕ)
37	3	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → 𝐼 ∈ ℕ)
38	5	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → 𝐼 ≤ 𝑀)
39	22	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → 𝑋 ∈ (1...𝑀))
40		simp2 1135	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → ¬ 𝑋 = 𝐼)
41		simp3 1136	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → ¬ 𝑋 < 𝐼)
42		metakunt31.9	. . . . . . 7 ⊢ 𝐻 = if(𝐼 ≤ (𝑋 − 𝐼), 1, 0)
43	36, 37, 38, 39, 7, 9, 40, 41, 8, 42	metakunt30 39661	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = ((𝑋 − 𝐼) + 𝐻))
44	40	iffalsed 4424	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))) = if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻)))
45	41	iffalsed 4424	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻)) = ((𝑋 − 𝐼) + 𝐻))
46	44, 45	eqtrd 2794	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))) = ((𝑋 − 𝐼) + 𝐻))
47	46	eqcomd 2765	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → ((𝑋 − 𝐼) + 𝐻) = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))))
48	43, 47	eqtrd 2794	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))))
49	16	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))) = 𝑅)
50	48, 49	eqtrd 2794	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅)
51	50	3expa 1116	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝐼) ∧ ¬ 𝑋 < 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅)
52	35, 51	pm2.61dan 813	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅)
53	18, 52	pm2.61dan 813	1 ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 ifcif 4413 class class class wbr 5025 ↦ cmpt 5105 ‘cfv 6328 (class class class)co 7143 0cc0 10560 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-rp 12416 df-fz 12925

This theorem is referenced by: metakunt33 39664

W3C validator