Theorem ex-fpar 30718

Description: Formalized example provided in the comment for fpar 8099. (Contributed by AV, 3-Jan-2024.)

Hypotheses

Ref	Expression
ex-fpar.h	⊢ 𝐻 = ((◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))
ex-fpar.a	⊢ 𝐴 = (0[,)+∞)
ex-fpar.b	⊢ 𝐵 = ℝ
ex-fpar.f	⊢ 𝐹 = (√ ↾ 𝐴)
ex-fpar.g	⊢ 𝐺 = (sin ↾ 𝐵)

Assertion

Ref	Expression
ex-fpar	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋( + ∘ 𝐻)𝑌) = ((√‘𝑋) + (sin‘𝑌)))

Proof of Theorem ex-fpar

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7403	. 2 ⊢ (𝑋( + ∘ 𝐻)𝑌) = (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩)
2		sqrtf 15403	. . . . . . . . 9 ⊢ √:ℂ⟶ℂ
3		ffn 6695	. . . . . . . . 9 ⊢ (√:ℂ⟶ℂ → √ Fn ℂ)
4	2, 3	ax-mp 5	. . . . . . . 8 ⊢ √ Fn ℂ
5		rge0ssre 13471	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ
6		ax-resscn 11145	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
7	5, 6	sstri 3948	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℂ
8		fnssres 6648	. . . . . . . . 9 ⊢ ((√ Fn ℂ ∧ (0[,)+∞) ⊆ ℂ) → (√ ↾ (0[,)+∞)) Fn (0[,)+∞))
9		ex-fpar.a	. . . . . . . . . . 11 ⊢ 𝐴 = (0[,)+∞)
10	9	reseq2i 5965	. . . . . . . . . 10 ⊢ (√ ↾ 𝐴) = (√ ↾ (0[,)+∞))
11	10	fneq1i 6622	. . . . . . . . 9 ⊢ ((√ ↾ 𝐴) Fn (0[,)+∞) ↔ (√ ↾ (0[,)+∞)) Fn (0[,)+∞))
12	8, 11	sylibr 237	. . . . . . . 8 ⊢ ((√ Fn ℂ ∧ (0[,)+∞) ⊆ ℂ) → (√ ↾ 𝐴) Fn (0[,)+∞))
13	4, 7, 12	mp2an 704	. . . . . . 7 ⊢ (√ ↾ 𝐴) Fn (0[,)+∞)
14		ex-fpar.f	. . . . . . . 8 ⊢ 𝐹 = (√ ↾ 𝐴)
15		id 23	. . . . . . . . 9 ⊢ (𝐹 = (√ ↾ 𝐴) → 𝐹 = (√ ↾ 𝐴))
16	9	a1i 11	. . . . . . . . 9 ⊢ (𝐹 = (√ ↾ 𝐴) → 𝐴 = (0[,)+∞))
17	15, 16	fneq12d 6620	. . . . . . . 8 ⊢ (𝐹 = (√ ↾ 𝐴) → (𝐹 Fn 𝐴 ↔ (√ ↾ 𝐴) Fn (0[,)+∞)))
18	14, 17	ax-mp 5	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 ↔ (√ ↾ 𝐴) Fn (0[,)+∞))
19	13, 18	mpbir 234	. . . . . 6 ⊢ 𝐹 Fn 𝐴
20		sinf 16168	. . . . . . . . 9 ⊢ sin:ℂ⟶ℂ
21		ffn 6695	. . . . . . . . 9 ⊢ (sin:ℂ⟶ℂ → sin Fn ℂ)
22	20, 21	ax-mp 5	. . . . . . . 8 ⊢ sin Fn ℂ
23		fnssres 6648	. . . . . . . . 9 ⊢ ((sin Fn ℂ ∧ ℝ ⊆ ℂ) → (sin ↾ ℝ) Fn ℝ)
24		ex-fpar.b	. . . . . . . . . . 11 ⊢ 𝐵 = ℝ
25	24	reseq2i 5965	. . . . . . . . . 10 ⊢ (sin ↾ 𝐵) = (sin ↾ ℝ)
26	25	fneq1i 6622	. . . . . . . . 9 ⊢ ((sin ↾ 𝐵) Fn ℝ ↔ (sin ↾ ℝ) Fn ℝ)
27	23, 26	sylibr 237	. . . . . . . 8 ⊢ ((sin Fn ℂ ∧ ℝ ⊆ ℂ) → (sin ↾ 𝐵) Fn ℝ)
28	22, 6, 27	mp2an 704	. . . . . . 7 ⊢ (sin ↾ 𝐵) Fn ℝ
29		ex-fpar.g	. . . . . . . 8 ⊢ 𝐺 = (sin ↾ 𝐵)
30		id 23	. . . . . . . . 9 ⊢ (𝐺 = (sin ↾ 𝐵) → 𝐺 = (sin ↾ 𝐵))
31	24	a1i 11	. . . . . . . . 9 ⊢ (𝐺 = (sin ↾ 𝐵) → 𝐵 = ℝ)
32	30, 31	fneq12d 6620	. . . . . . . 8 ⊢ (𝐺 = (sin ↾ 𝐵) → (𝐺 Fn 𝐵 ↔ (sin ↾ 𝐵) Fn ℝ))
33	29, 32	ax-mp 5	. . . . . . 7 ⊢ (𝐺 Fn 𝐵 ↔ (sin ↾ 𝐵) Fn ℝ)
34	28, 33	mpbir 234	. . . . . 6 ⊢ 𝐺 Fn 𝐵
35		ex-fpar.h	. . . . . . 7 ⊢ 𝐻 = ((◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))
36	35	fpar 8099	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩))
37	19, 34, 36	mp2an 704	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
38		opex 5435	. . . . 5 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V
39	37, 38	fnmpoi 8055	. . . 4 ⊢ 𝐻 Fn (𝐴 × 𝐵)
40		opelxpi 5688	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
41		fvco2 6968	. . . 4 ⊢ ((𝐻 Fn (𝐴 × 𝐵) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵)) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)))
42	39, 40, 41	sylancr 598	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)))
43		simpl 487	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐴)
44		simpr 489	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
45	37, 43, 44	fvproj 8118	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)
46	45	fveq2d 6875	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)) = ( + ‘⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩))
47		df-ov 7403	. . . 4 ⊢ ((𝐹‘𝑋) + (𝐺‘𝑌)) = ( + ‘⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)
48	14	fveq1i 6872	. . . . . 6 ⊢ (𝐹‘𝑋) = ((√ ↾ 𝐴)‘𝑋)
49		fvres 6890	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → ((√ ↾ 𝐴)‘𝑋) = (√‘𝑋))
50	48, 49	eqtrid 2812	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝐹‘𝑋) = (√‘𝑋))
51	29	fveq1i 6872	. . . . . 6 ⊢ (𝐺‘𝑌) = ((sin ↾ 𝐵)‘𝑌)
52		fvres 6890	. . . . . 6 ⊢ (𝑌 ∈ 𝐵 → ((sin ↾ 𝐵)‘𝑌) = (sin‘𝑌))
53	51, 52	eqtrid 2812	. . . . 5 ⊢ (𝑌 ∈ 𝐵 → (𝐺‘𝑌) = (sin‘𝑌))
54	50, 53	oveqan12d 7419	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ((𝐹‘𝑋) + (𝐺‘𝑌)) = ((√‘𝑋) + (sin‘𝑌)))
55	47, 54	eqtr3id 2814	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ( + ‘⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩) = ((√‘𝑋) + (sin‘𝑌)))
56	42, 46, 55	3eqtrd 2804	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ((√‘𝑋) + (sin‘𝑌)))
57	1, 56	eqtrid 2812	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋( + ∘ 𝐻)𝑌) = ((√‘𝑋) + (sin‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∩ cin 3906   ⊆ wss 3907  ⟨cop 4591   × cxp 5649  ^◡ccnv 5650   ↾ cres 5653   ∘ ccom 5655   Fn wfn 6520  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  1^st c1st 7972  2^nd c2nd 7973  ℂcc 11086  ℝcr 11087  0cc0 11088   + caddc 11091  +∞cpnf 11228  [,)cico 13362  √csqrt 15272  sincsin 16105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-inf2 9598  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-pm 8815  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-sup 9390  df-inf 9391  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12222  df-2 12291  df-3 12292  df-n0 12493  df-z 12580  df-uz 12851  df-rp 13005  df-ico 13366  df-fz 13524  df-fzo 13671  df-fl 13813  df-seq 14026  df-exp 14086  df-fac 14298  df-hash 14355  df-shft 15092  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15510  df-clim 15527  df-rlim 15528  df-sum 15726  df-ef 16109  df-sin 16111

This theorem is referenced by: (None)