Theorem oprpiece1res2 25273

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 11367	. . . . 5 ⊢ 𝐴 ∈ ℝ*
4		oprpiece1.2	. . . . . 6 ⊢ 𝐵 ∈ ℝ
5	4	rexri 11367	. . . . 5 ⊢ 𝐵 ∈ ℝ*
6		oprpiece1.3	. . . . 5 ⊢ 𝐴 ≤ 𝐵
7		ubicc2 13596	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
8	3, 5, 6, 7	mp3an 1490	. . . 4 ⊢ 𝐵 ∈ (𝐴[,]𝐵)
9		iccss2 13548	. . . 4 ⊢ ((𝐾 ∈ (𝐴[,]𝐵) ∧ 𝐵 ∈ (𝐴[,]𝐵)) → (𝐾[,]𝐵) ⊆ (𝐴[,]𝐵))
10	1, 8, 9	mp2an 705	. . 3 ⊢ (𝐾[,]𝐵) ⊆ (𝐴[,]𝐵)
11		ssid 3930	. . 3 ⊢ 𝐶 ⊆ 𝐶
12		resmpo 7540	. . 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 705	. 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 5965	. 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 740	. . . . . 6 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑃 = 𝑄)
19		simpr 490	. . . . . . . 8 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑥 ≤ 𝐾)
20	2, 4	elicc2i 13543	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (𝐴[,]𝐵) ↔ (𝐾 ∈ ℝ ∧ 𝐴 ≤ 𝐾 ∧ 𝐾 ≤ 𝐵))
21	20	simp1bi 1163	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ (𝐴[,]𝐵) → 𝐾 ∈ ℝ)
22	1, 21	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝐾 ∈ ℝ
23	22, 4	elicc2i 13543	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐾[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐾 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
24	23	simp2bi 1164	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐾[,]𝐵) → 𝐾 ≤ 𝑥)
25	24	ad2antrr 739	. . . . . . . 8 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝐾 ≤ 𝑥)
26	23	simp1bi 1163	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐾[,]𝐵) → 𝑥 ∈ ℝ)
27	26	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑥 ∈ ℝ)
28		letri3 11395	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (𝑥 = 𝐾 ↔ (𝑥 ≤ 𝐾 ∧ 𝐾 ≤ 𝑥)))
29	27, 22, 28	sylancl 598	. . . . . . . 8 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → (𝑥 = 𝐾 ↔ (𝑥 ≤ 𝐾 ∧ 𝐾 ≤ 𝑥)))
30	19, 25, 29	mpbir2and 726	. . . . . . 7 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑥 = 𝐾)
31		oprpiece1.9	. . . . . . 7 ⊢ (𝑥 = 𝐾 → 𝑅 = 𝑃)
32	30, 31	syl 18	. . . . . 6 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑅 = 𝑃)
33		oprpiece1.10	. . . . . . 7 ⊢ (𝑥 = 𝐾 → 𝑆 = 𝑄)
34	30, 33	syl 18	. . . . . 6 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑆 = 𝑄)
35	18, 32, 34	3eqtr4d 2805	. . . . 5 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑅 = 𝑆)
36		eqidd 2761	. . . . 5 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ ¬ 𝑥 ≤ 𝐾) → 𝑆 = 𝑆)
37	35, 36	ifeqda 4518	. . . 4 ⊢ ((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) → if(𝑥 ≤ 𝐾, 𝑅, 𝑆) = 𝑆)
38	37	mpoeq3ia 7498	. . 3 ⊢ (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ 𝑆)
39	16, 38	eqtr4i 2786	. 2 ⊢ 𝐺 = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆))
40	13, 15, 39	3eqtr4i 2793	1 ⊢ (𝐹 ↾ ((𝐾[,]𝐵) × 𝐶)) = 𝐺

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895  ifcif 4481   class class class wbr 5102   × cxp 5648   ↾ cres 5652  (class class class)co 7420   ∈ cmpo 7422  ℝcr 11199  ℝ^*cxr 11342   ≤ cle 11344  [,]cicc 13479

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-pre-lttri 11274  ax-pre-lttrn 11275

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-po 5558  df-so 5559  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-icc 13483

This theorem is used by: (None)