metakunt31 - Mathbox for metakunt

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Theorem metakunt31 41743

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 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → 𝑀 ∈ ℕ)
3		metakunt31.2	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℕ)
4	3	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → 𝐼 ∈ ℕ)
5		metakunt31.3	. . . . . 6 ⊢ (𝜑 → 𝐼 ≤ 𝑀)
6	5	adantr 479	. . . . 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 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → 𝑋 = 𝐼)
11	2, 4, 6, 7, 8, 9, 10	metakunt26 41738	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑋)
12	10	iftrued 4532	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))) = 𝑋)
13	12	eqcomd 2731	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → 𝑋 = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))))
14	11, 13	eqtrd 2765	. . 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 2734	. . . 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 2765	. 2 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅)
19	1	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → 𝑀 ∈ ℕ)
20	3	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → 𝐼 ∈ ℕ)
21	5	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → 𝐼 ≤ 𝑀)
22		metakunt31.4	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (1...𝑀))
23	22	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → 𝑋 ∈ (1...𝑀))
24		simp2 1134	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → ¬ 𝑋 = 𝐼)
25		simp3 1135	. . . . . . 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 41741	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = ((𝑋 + (𝑀 − 𝐼)) + 𝐺))
28	24	iffalsed 4535	. . . . . . . 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 4532	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻)) = ((𝑋 + (𝑀 − 𝐼)) + 𝐺))
30	28, 29	eqtrd 2765	. . . . . . 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 2731	. . . . . 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 2765	. . . . 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 2765	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ 𝑋 < 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅)
35	34	3expa 1115	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝐼) ∧ 𝑋 < 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅)
36	1	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → 𝑀 ∈ ℕ)
37	3	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → 𝐼 ∈ ℕ)
38	5	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → 𝐼 ≤ 𝑀)
39	22	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → 𝑋 ∈ (1...𝑀))
40		simp2 1134	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → ¬ 𝑋 = 𝐼)
41		simp3 1135	. . . . . . 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 41742	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = ((𝑋 − 𝐼) + 𝐻))
44	40	iffalsed 4535	. . . . . . . 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 4535	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻)) = ((𝑋 − 𝐼) + 𝐻))
46	44, 45	eqtrd 2765	. . . . . . 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 2731	. . . . . 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 2765	. . . . 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 2765	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼 ∧ ¬ 𝑋 < 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅)
51	50	3expa 1115	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝐼) ∧ ¬ 𝑋 < 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅)
52	35, 51	pm2.61dan 811	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐼) → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅)
53	18, 52	pm2.61dan 811	1 ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ifcif 4524 class class class wbr 5143 ↦ cmpt 5226 ‘cfv 6543 (class class class)co 7416 0cc0 11138 1c1 11139 + caddc 11141 < clt 11278 ≤ cle 11279 − cmin 11474 ℕcn 12242 ...cfz 13516

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-fz 13517

This theorem is referenced by: metakunt33 41745

W3C validator