Theorem oprpiece1res2 23249

Description: Restriction to the second part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)

Hypotheses

Ref	Expression
oprpiece1.1	⊢ 𝐴 ∈ ℝ
oprpiece1.2	⊢ 𝐵 ∈ ℝ
oprpiece1.3	⊢ 𝐴 ≤ 𝐵
oprpiece1.4	⊢ 𝑅 ∈ V
oprpiece1.5	⊢ 𝑆 ∈ V
oprpiece1.6	⊢ 𝐾 ∈ (𝐴[,]𝐵)
oprpiece1.7	⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆))
oprpiece1.9	⊢ (𝑥 = 𝐾 → 𝑅 = 𝑃)
oprpiece1.10	⊢ (𝑥 = 𝐾 → 𝑆 = 𝑄)
oprpiece1.11	⊢ (𝑦 ∈ 𝐶 → 𝑃 = 𝑄)
oprpiece1.12	⊢ 𝐺 = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ 𝑆)

Assertion

Ref	Expression
oprpiece1res2	⊢ (𝐹 ↾ ((𝐾[,]𝐵) × 𝐶)) = 𝐺

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐾,𝑦   𝑥,𝑃   𝑥,𝑄

Allowed substitution hints:   𝑃(𝑦)   𝑄(𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem oprpiece1res2

Step	Hyp	Ref	Expression
1		oprpiece1.6	. . . 4 ⊢ 𝐾 ∈ (𝐴[,]𝐵)
2		oprpiece1.1	. . . . . 6 ⊢ 𝐴 ∈ ℝ
3	2	rexri 10491	. . . . 5 ⊢ 𝐴 ∈ ℝ*
4		oprpiece1.2	. . . . . 6 ⊢ 𝐵 ∈ ℝ
5	4	rexri 10491	. . . . 5 ⊢ 𝐵 ∈ ℝ*
6		oprpiece1.3	. . . . 5 ⊢ 𝐴 ≤ 𝐵
7		ubicc2 12662	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
8	3, 5, 6, 7	mp3an 1440	. . . 4 ⊢ 𝐵 ∈ (𝐴[,]𝐵)
9		iccss2 12616	. . . 4 ⊢ ((𝐾 ∈ (𝐴[,]𝐵) ∧ 𝐵 ∈ (𝐴[,]𝐵)) → (𝐾[,]𝐵) ⊆ (𝐴[,]𝐵))
10	1, 8, 9	mp2an 679	. . 3 ⊢ (𝐾[,]𝐵) ⊆ (𝐴[,]𝐵)
11		ssid 3875	. . 3 ⊢ 𝐶 ⊆ 𝐶
12		resmpo 7082	. . 3 ⊢ (((𝐾[,]𝐵) ⊆ (𝐴[,]𝐵) ∧ 𝐶 ⊆ 𝐶) → ((𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) ↾ ((𝐾[,]𝐵) × 𝐶)) = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)))
13	10, 11, 12	mp2an 679	. 2 ⊢ ((𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) ↾ ((𝐾[,]𝐵) × 𝐶)) = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆))
14		oprpiece1.7	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆))
15	14	reseq1i 5684	. 2 ⊢ (𝐹 ↾ ((𝐾[,]𝐵) × 𝐶)) = ((𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) ↾ ((𝐾[,]𝐵) × 𝐶))
16		oprpiece1.12	. . 3 ⊢ 𝐺 = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ 𝑆)
17		oprpiece1.11	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → 𝑃 = 𝑄)
18	17	ad2antlr 714	. . . . . 6 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑃 = 𝑄)
19		simpr 477	. . . . . . . 8 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑥 ≤ 𝐾)
20	2, 4	elicc2i 12611	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (𝐴[,]𝐵) ↔ (𝐾 ∈ ℝ ∧ 𝐴 ≤ 𝐾 ∧ 𝐾 ≤ 𝐵))
21	20	simp1bi 1125	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ (𝐴[,]𝐵) → 𝐾 ∈ ℝ)
22	1, 21	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝐾 ∈ ℝ
23	22, 4	elicc2i 12611	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐾[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐾 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
24	23	simp2bi 1126	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐾[,]𝐵) → 𝐾 ≤ 𝑥)
25	24	ad2antrr 713	. . . . . . . 8 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝐾 ≤ 𝑥)
26	23	simp1bi 1125	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐾[,]𝐵) → 𝑥 ∈ ℝ)
27	26	ad2antrr 713	. . . . . . . . 9 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑥 ∈ ℝ)
28		letri3 10518	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (𝑥 = 𝐾 ↔ (𝑥 ≤ 𝐾 ∧ 𝐾 ≤ 𝑥)))
29	27, 22, 28	sylancl 577	. . . . . . . 8 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → (𝑥 = 𝐾 ↔ (𝑥 ≤ 𝐾 ∧ 𝐾 ≤ 𝑥)))
30	19, 25, 29	mpbir2and 700	. . . . . . 7 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑥 = 𝐾)
31		oprpiece1.9	. . . . . . 7 ⊢ (𝑥 = 𝐾 → 𝑅 = 𝑃)
32	30, 31	syl 17	. . . . . 6 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑅 = 𝑃)
33		oprpiece1.10	. . . . . . 7 ⊢ (𝑥 = 𝐾 → 𝑆 = 𝑄)
34	30, 33	syl 17	. . . . . 6 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑆 = 𝑄)
35	18, 32, 34	3eqtr4d 2818	. . . . 5 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑅 = 𝑆)
36		eqidd 2773	. . . . 5 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ ¬ 𝑥 ≤ 𝐾) → 𝑆 = 𝑆)
37	35, 36	ifeqda 4379	. . . 4 ⊢ ((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) → if(𝑥 ≤ 𝐾, 𝑅, 𝑆) = 𝑆)
38	37	mpoeq3ia 7044	. . 3 ⊢ (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ 𝑆)
39	16, 38	eqtr4i 2799	. 2 ⊢ 𝐺 = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆))
40	13, 15, 39	3eqtr4i 2806	1 ⊢ (𝐹 ↾ ((𝐾[,]𝐵) × 𝐶)) = 𝐺

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2048  Vcvv 3409   ⊆ wss 3825  ifcif 4344   class class class wbr 4923   × cxp 5398   ↾ cres 5402  (class class class)co 6970   ∈ cmpo 6972  ℝcr 10326  ℝ^*cxr 10465   ≤ cle 10467  [,]cicc 12550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-pre-lttri 10401  ax-pre-lttrn 10402

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-po 5319  df-so 5320  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7494  df-2nd 7495  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-icc 12554

This theorem is referenced by: (None)