Theorem oprpiece1res2 24906

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 11190	. . . . 5 ⊢ 𝐴 ∈ ℝ*
4		oprpiece1.2	. . . . . 6 ⊢ 𝐵 ∈ ℝ
5	4	rexri 11190	. . . . 5 ⊢ 𝐵 ∈ ℝ*
6		oprpiece1.3	. . . . 5 ⊢ 𝐴 ≤ 𝐵
7		ubicc2 13381	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
8	3, 5, 6, 7	mp3an 1463	. . . 4 ⊢ 𝐵 ∈ (𝐴[,]𝐵)
9		iccss2 13333	. . . 4 ⊢ ((𝐾 ∈ (𝐴[,]𝐵) ∧ 𝐵 ∈ (𝐴[,]𝐵)) → (𝐾[,]𝐵) ⊆ (𝐴[,]𝐵))
10	1, 8, 9	mp2an 692	. . 3 ⊢ (𝐾[,]𝐵) ⊆ (𝐴[,]𝐵)
11		ssid 3956	. . 3 ⊢ 𝐶 ⊆ 𝐶
12		resmpo 7478	. . 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 692	. 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 5934	. 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 727	. . . . . 6 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑃 = 𝑄)
19		simpr 484	. . . . . . . 8 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑥 ≤ 𝐾)
20	2, 4	elicc2i 13328	. . . . . . . . . . . . 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 13328	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐾[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐾 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
24	23	simp2bi 1146	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐾[,]𝐵) → 𝐾 ≤ 𝑥)
25	24	ad2antrr 726	. . . . . . . 8 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝐾 ≤ 𝑥)
26	23	simp1bi 1145	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐾[,]𝐵) → 𝑥 ∈ ℝ)
27	26	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑥 ∈ ℝ)
28		letri3 11218	. . . . . . . . 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 713	. . . . . . 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 2781	. . . . 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 4516	. . . 4 ⊢ ((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) → if(𝑥 ≤ 𝐾, 𝑅, 𝑆) = 𝑆)
38	37	mpoeq3ia 7436	. . 3 ⊢ (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ 𝑆)
39	16, 38	eqtr4i 2762	. 2 ⊢ 𝐺 = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆))
40	13, 15, 39	3eqtr4i 2769	1 ⊢ (𝐹 ↾ ((𝐾[,]𝐵) × 𝐶)) = 𝐺

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ⊆ wss 3901  ifcif 4479   class class class wbr 5098   × cxp 5622   ↾ cres 5626  (class class class)co 7358   ∈ cmpo 7360  ℝcr 11025  ℝ^*cxr 11165   ≤ cle 11167  [,]cicc 13264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-pre-lttri 11100  ax-pre-lttrn 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-icc 13268

This theorem is referenced by: (None)