Theorem oprpiece1res2 24315

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 11213	. . . . 5 ⊢ 𝐴 ∈ ℝ*
4		oprpiece1.2	. . . . . 6 ⊢ 𝐵 ∈ ℝ
5	4	rexri 11213	. . . . 5 ⊢ 𝐵 ∈ ℝ*
6		oprpiece1.3	. . . . 5 ⊢ 𝐴 ≤ 𝐵
7		ubicc2 13382	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
8	3, 5, 6, 7	mp3an 1461	. . . 4 ⊢ 𝐵 ∈ (𝐴[,]𝐵)
9		iccss2 13335	. . . 4 ⊢ ((𝐾 ∈ (𝐴[,]𝐵) ∧ 𝐵 ∈ (𝐴[,]𝐵)) → (𝐾[,]𝐵) ⊆ (𝐴[,]𝐵))
10	1, 8, 9	mp2an 690	. . 3 ⊢ (𝐾[,]𝐵) ⊆ (𝐴[,]𝐵)
11		ssid 3966	. . 3 ⊢ 𝐶 ⊆ 𝐶
12		resmpo 7476	. . 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 690	. 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 5933	. 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 725	. . . . . 6 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑃 = 𝑄)
19		simpr 485	. . . . . . . 8 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑥 ≤ 𝐾)
20	2, 4	elicc2i 13330	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (𝐴[,]𝐵) ↔ (𝐾 ∈ ℝ ∧ 𝐴 ≤ 𝐾 ∧ 𝐾 ≤ 𝐵))
21	20	simp1bi 1145	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ (𝐴[,]𝐵) → 𝐾 ∈ ℝ)
22	1, 21	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝐾 ∈ ℝ
23	22, 4	elicc2i 13330	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐾[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐾 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
24	23	simp2bi 1146	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐾[,]𝐵) → 𝐾 ≤ 𝑥)
25	24	ad2antrr 724	. . . . . . . 8 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝐾 ≤ 𝑥)
26	23	simp1bi 1145	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐾[,]𝐵) → 𝑥 ∈ ℝ)
27	26	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑥 ∈ ℝ)
28		letri3 11240	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (𝑥 = 𝐾 ↔ (𝑥 ≤ 𝐾 ∧ 𝐾 ≤ 𝑥)))
29	27, 22, 28	sylancl 586	. . . . . . . 8 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → (𝑥 = 𝐾 ↔ (𝑥 ≤ 𝐾 ∧ 𝐾 ≤ 𝑥)))
30	19, 25, 29	mpbir2and 711	. . . . . . 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 2786	. . . . 5 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑅 = 𝑆)
36		eqidd 2737	. . . . 5 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ ¬ 𝑥 ≤ 𝐾) → 𝑆 = 𝑆)
37	35, 36	ifeqda 4522	. . . 4 ⊢ ((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) → if(𝑥 ≤ 𝐾, 𝑅, 𝑆) = 𝑆)
38	37	mpoeq3ia 7435	. . 3 ⊢ (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ 𝑆)
39	16, 38	eqtr4i 2767	. 2 ⊢ 𝐺 = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆))
40	13, 15, 39	3eqtr4i 2774	1 ⊢ (𝐹 ↾ ((𝐾[,]𝐵) × 𝐶)) = 𝐺

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3445   ⊆ wss 3910  ifcif 4486   class class class wbr 5105   × cxp 5631   ↾ cres 5635  (class class class)co 7357   ∈ cmpo 7359  ℝcr 11050  ℝ^*cxr 11188   ≤ cle 11190  [,]cicc 13267

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-pre-lttri 11125  ax-pre-lttrn 11126

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-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  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-id 5531  df-po 5545  df-so 5546  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-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  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-icc 13271

This theorem is referenced by: (None)