Theorem offval 7665

Description: Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)

Hypotheses

Ref	Expression
offval.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
offval.2	⊢ (𝜑 → 𝐺 Fn 𝐵)
offval.3	⊢ (𝜑 → 𝐴 ∈ 𝑉)
offval.4	⊢ (𝜑 → 𝐵 ∈ 𝑊)
offval.5	⊢ (𝐴 ∩ 𝐵) = 𝑆
offval.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
offval.7	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷)

Assertion

Ref	Expression
offval	⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ (𝐶𝑅𝐷)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥   𝑥,𝑆   𝑥,𝑅

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem offval

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		offval.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		fnex 7197	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
4	1, 2, 3	syl2anc 593	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
5		offval.2	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵)
6		offval.4	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		fnex 7197	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑊) → 𝐺 ∈ V)
8	5, 6, 7	syl2anc 593	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
9	1	fndmd 6622	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐴)
10	5	fndmd 6622	. . . . . . 7 ⊢ (𝜑 → dom 𝐺 = 𝐵)
11	9, 10	ineq12d 4173	. . . . . 6 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
12		offval.5	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) = 𝑆
13	11, 12	eqtrdi 2812	. . . . 5 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
14	13	mpteq1d 5189	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
15		inex1g 5274	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
16	12, 15	eqeltrrid 2866	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝑆 ∈ V)
17		mptexg 7201	. . . . 5 ⊢ (𝑆 ∈ V → (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V)
18	2, 16, 17	3syl 18	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V)
19	14, 18	eqeltrd 2861	. . 3 ⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V)
20		dmeq 5877	. . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
21		dmeq 5877	. . . . . 6 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
22	20, 21	ineqan12d 4174	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
23		fveq1 6862	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
24		fveq1 6862	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥))
25	23, 24	oveqan12d 7411	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) = ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
26	22, 25	mpteq12dv 5186	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
27		df-of 7656	. . . 4 ⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
28	26, 27	ovmpoga 7546	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V ∧ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V) → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
29	4, 8, 19, 28	syl3anc 1389	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
30	12	eleq2i 2853	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ 𝑥 ∈ 𝑆)
31		elin 3920	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
32	30, 31	bitr3i 279	. . . 4 ⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
33		offval.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
34	33	adantrr 727	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → (𝐹‘𝑥) = 𝐶)
35		offval.7	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷)
36	35	adantrl 726	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → (𝐺‘𝑥) = 𝐷)
37	34, 36	oveq12d 7410	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝐶𝑅𝐷))
38	32, 37	sylan2b 603	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝐶𝑅𝐷))
39	38	mpteq2dva 5192	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ (𝐶𝑅𝐷)))
40	29, 14, 39	3eqtrd 2800	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ (𝐶𝑅𝐷)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ∩ cin 3903   ↦ cmpt 5180  dom cdm 5645   Fn wfn 6512  ‘cfv 6517  (class class class)co 7392   ∘_f cof 7654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-of 7656

This theorem is referenced by:  ofval  7667  offn  7669  offval2f  7671  off  7674  ofres  7675  offval2  7676  coof  7680  ofco  7681  offveqb  7683  suppssof1  8174  o1rlimmul  15629  frlmipval  21811  frlmphllem  21812  frlmphl  21813  gsumbagdiaglem  21963  psrascl  22010  evlslem1  22115  evlsvvval  22126  mhpmulcl  22194  psdmplcl  22207  psdadd  22208  psdmul  22211  psrplusgpropd  22277  evls1fpws  22412  mat1dimscm  22515  rrxcph  25434  rrxds  25435  mbfadd  25703  mbfsub  25704  mbfmullem2  25766  mbfmul  25768  bddmulibl  25881  dvcmulf  25987  ofrn2  32792  off2  32793  ofresid  32794  islinds5  33514  ellspds  33515  ply1gsumz  33756  extdgfialglem2  33951  ofcof  34365  plymul02  34804  signsplypnf  34808  signsply0  34809  matunitlindflem1  38079  matunitlindflem2  38080  poimirlem4  38087  poimirlem16  38099  poimirlem19  38102  poimirlem28  38111  broucube  38117  itg2addnc  38137  ftc1anclem8  38163  dflinc2  48996  fdivmpt  49126