Theorem ex-fpar 30424

Description: Formalized example provided in the comment for fpar 8056. (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 7356	. 2 ⊢ (𝑋( + ∘ 𝐻)𝑌) = (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩)
2		sqrtf 15289	. . . . . . . . 9 ⊢ √:ℂ⟶ℂ
3		ffn 6656	. . . . . . . . 9 ⊢ (√:ℂ⟶ℂ → √ Fn ℂ)
4	2, 3	ax-mp 5	. . . . . . . 8 ⊢ √ Fn ℂ
5		rge0ssre 13377	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ
6		ax-resscn 11085	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
7	5, 6	sstri 3947	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℂ
8		fnssres 6609	. . . . . . . . 9 ⊢ ((√ Fn ℂ ∧ (0[,)+∞) ⊆ ℂ) → (√ ↾ (0[,)+∞)) Fn (0[,)+∞))
9		ex-fpar.a	. . . . . . . . . . 11 ⊢ 𝐴 = (0[,)+∞)
10	9	reseq2i 5931	. . . . . . . . . 10 ⊢ (√ ↾ 𝐴) = (√ ↾ (0[,)+∞))
11	10	fneq1i 6583	. . . . . . . . 9 ⊢ ((√ ↾ 𝐴) Fn (0[,)+∞) ↔ (√ ↾ (0[,)+∞)) Fn (0[,)+∞))
12	8, 11	sylibr 234	. . . . . . . 8 ⊢ ((√ Fn ℂ ∧ (0[,)+∞) ⊆ ℂ) → (√ ↾ 𝐴) Fn (0[,)+∞))
13	4, 7, 12	mp2an 692	. . . . . . 7 ⊢ (√ ↾ 𝐴) Fn (0[,)+∞)
14		ex-fpar.f	. . . . . . . 8 ⊢ 𝐹 = (√ ↾ 𝐴)
15		id 22	. . . . . . . . 9 ⊢ (𝐹 = (√ ↾ 𝐴) → 𝐹 = (√ ↾ 𝐴))
16	9	a1i 11	. . . . . . . . 9 ⊢ (𝐹 = (√ ↾ 𝐴) → 𝐴 = (0[,)+∞))
17	15, 16	fneq12d 6581	. . . . . . . 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 231	. . . . . 6 ⊢ 𝐹 Fn 𝐴
20		sinf 16051	. . . . . . . . 9 ⊢ sin:ℂ⟶ℂ
21		ffn 6656	. . . . . . . . 9 ⊢ (sin:ℂ⟶ℂ → sin Fn ℂ)
22	20, 21	ax-mp 5	. . . . . . . 8 ⊢ sin Fn ℂ
23		fnssres 6609	. . . . . . . . 9 ⊢ ((sin Fn ℂ ∧ ℝ ⊆ ℂ) → (sin ↾ ℝ) Fn ℝ)
24		ex-fpar.b	. . . . . . . . . . 11 ⊢ 𝐵 = ℝ
25	24	reseq2i 5931	. . . . . . . . . 10 ⊢ (sin ↾ 𝐵) = (sin ↾ ℝ)
26	25	fneq1i 6583	. . . . . . . . 9 ⊢ ((sin ↾ 𝐵) Fn ℝ ↔ (sin ↾ ℝ) Fn ℝ)
27	23, 26	sylibr 234	. . . . . . . 8 ⊢ ((sin Fn ℂ ∧ ℝ ⊆ ℂ) → (sin ↾ 𝐵) Fn ℝ)
28	22, 6, 27	mp2an 692	. . . . . . 7 ⊢ (sin ↾ 𝐵) Fn ℝ
29		ex-fpar.g	. . . . . . . 8 ⊢ 𝐺 = (sin ↾ 𝐵)
30		id 22	. . . . . . . . 9 ⊢ (𝐺 = (sin ↾ 𝐵) → 𝐺 = (sin ↾ 𝐵))
31	24	a1i 11	. . . . . . . . 9 ⊢ (𝐺 = (sin ↾ 𝐵) → 𝐵 = ℝ)
32	30, 31	fneq12d 6581	. . . . . . . 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 231	. . . . . 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 8056	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩))
37	19, 34, 36	mp2an 692	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
38		opex 5411	. . . . 5 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V
39	37, 38	fnmpoi 8012	. . . 4 ⊢ 𝐻 Fn (𝐴 × 𝐵)
40		opelxpi 5660	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
41		fvco2 6924	. . . 4 ⊢ ((𝐻 Fn (𝐴 × 𝐵) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵)) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)))
42	39, 40, 41	sylancr 587	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)))
43		simpl 482	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐴)
44		simpr 484	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
45	37, 43, 44	fvproj 8074	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)
46	45	fveq2d 6830	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)) = ( + ‘⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩))
47		df-ov 7356	. . . 4 ⊢ ((𝐹‘𝑋) + (𝐺‘𝑌)) = ( + ‘⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)
48	14	fveq1i 6827	. . . . . 6 ⊢ (𝐹‘𝑋) = ((√ ↾ 𝐴)‘𝑋)
49		fvres 6845	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → ((√ ↾ 𝐴)‘𝑋) = (√‘𝑋))
50	48, 49	eqtrid 2776	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝐹‘𝑋) = (√‘𝑋))
51	29	fveq1i 6827	. . . . . 6 ⊢ (𝐺‘𝑌) = ((sin ↾ 𝐵)‘𝑌)
52		fvres 6845	. . . . . 6 ⊢ (𝑌 ∈ 𝐵 → ((sin ↾ 𝐵)‘𝑌) = (sin‘𝑌))
53	51, 52	eqtrid 2776	. . . . 5 ⊢ (𝑌 ∈ 𝐵 → (𝐺‘𝑌) = (sin‘𝑌))
54	50, 53	oveqan12d 7372	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ((𝐹‘𝑋) + (𝐺‘𝑌)) = ((√‘𝑋) + (sin‘𝑌)))
55	47, 54	eqtr3id 2778	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ( + ‘⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩) = ((√‘𝑋) + (sin‘𝑌)))
56	42, 46, 55	3eqtrd 2768	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ((√‘𝑋) + (sin‘𝑌)))
57	1, 56	eqtrid 2776	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋( + ∘ 𝐻)𝑌) = ((√‘𝑋) + (sin‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438   ∩ cin 3904   ⊆ wss 3905  ⟨cop 4585   × cxp 5621  ^◡ccnv 5622   ↾ cres 5625   ∘ ccom 5627   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  1^st c1st 7929  2^nd c2nd 7930  ℂcc 11026  ℝcr 11027  0cc0 11028   + caddc 11031  +∞cpnf 11165  [,)cico 13268  √csqrt 15158  sincsin 15988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-pm 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9351  df-inf 9352  df-oi 9421  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-div 11796  df-nn 12147  df-2 12209  df-3 12210  df-n0 12403  df-z 12490  df-uz 12754  df-rp 12912  df-ico 13272  df-fz 13429  df-fzo 13576  df-fl 13714  df-seq 13927  df-exp 13987  df-fac 14199  df-hash 14256  df-shft 14992  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161  df-limsup 15396  df-clim 15413  df-rlim 15414  df-sum 15612  df-ef 15992  df-sin 15994

This theorem is referenced by: (None)