Theorem ex-fpar 30537

Description: Formalized example provided in the comment for fpar 8058. (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 7361	. 2 ⊢ (𝑋( + ∘ 𝐻)𝑌) = (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩)
2		sqrtf 15287	. . . . . . . . 9 ⊢ √:ℂ⟶ℂ
3		ffn 6662	. . . . . . . . 9 ⊢ (√:ℂ⟶ℂ → √ Fn ℂ)
4	2, 3	ax-mp 5	. . . . . . . 8 ⊢ √ Fn ℂ
5		rge0ssre 13372	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ
6		ax-resscn 11083	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
7	5, 6	sstri 3943	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℂ
8		fnssres 6615	. . . . . . . . 9 ⊢ ((√ Fn ℂ ∧ (0[,)+∞) ⊆ ℂ) → (√ ↾ (0[,)+∞)) Fn (0[,)+∞))
9		ex-fpar.a	. . . . . . . . . . 11 ⊢ 𝐴 = (0[,)+∞)
10	9	reseq2i 5935	. . . . . . . . . 10 ⊢ (√ ↾ 𝐴) = (√ ↾ (0[,)+∞))
11	10	fneq1i 6589	. . . . . . . . 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 6587	. . . . . . . 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 16049	. . . . . . . . 9 ⊢ sin:ℂ⟶ℂ
21		ffn 6662	. . . . . . . . 9 ⊢ (sin:ℂ⟶ℂ → sin Fn ℂ)
22	20, 21	ax-mp 5	. . . . . . . 8 ⊢ sin Fn ℂ
23		fnssres 6615	. . . . . . . . 9 ⊢ ((sin Fn ℂ ∧ ℝ ⊆ ℂ) → (sin ↾ ℝ) Fn ℝ)
24		ex-fpar.b	. . . . . . . . . . 11 ⊢ 𝐵 = ℝ
25	24	reseq2i 5935	. . . . . . . . . 10 ⊢ (sin ↾ 𝐵) = (sin ↾ ℝ)
26	25	fneq1i 6589	. . . . . . . . 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 6587	. . . . . . . 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 8058	. . . . . 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 5412	. . . . 5 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V
39	37, 38	fnmpoi 8014	. . . 4 ⊢ 𝐻 Fn (𝐴 × 𝐵)
40		opelxpi 5661	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
41		fvco2 6931	. . . 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 8076	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)
46	45	fveq2d 6838	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)) = ( + ‘⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩))
47		df-ov 7361	. . . 4 ⊢ ((𝐹‘𝑋) + (𝐺‘𝑌)) = ( + ‘⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)
48	14	fveq1i 6835	. . . . . 6 ⊢ (𝐹‘𝑋) = ((√ ↾ 𝐴)‘𝑋)
49		fvres 6853	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → ((√ ↾ 𝐴)‘𝑋) = (√‘𝑋))
50	48, 49	eqtrid 2783	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝐹‘𝑋) = (√‘𝑋))
51	29	fveq1i 6835	. . . . . 6 ⊢ (𝐺‘𝑌) = ((sin ↾ 𝐵)‘𝑌)
52		fvres 6853	. . . . . 6 ⊢ (𝑌 ∈ 𝐵 → ((sin ↾ 𝐵)‘𝑌) = (sin‘𝑌))
53	51, 52	eqtrid 2783	. . . . 5 ⊢ (𝑌 ∈ 𝐵 → (𝐺‘𝑌) = (sin‘𝑌))
54	50, 53	oveqan12d 7377	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ((𝐹‘𝑋) + (𝐺‘𝑌)) = ((√‘𝑋) + (sin‘𝑌)))
55	47, 54	eqtr3id 2785	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ( + ‘⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩) = ((√‘𝑋) + (sin‘𝑌)))
56	42, 46, 55	3eqtrd 2775	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ((√‘𝑋) + (sin‘𝑌)))
57	1, 56	eqtrid 2783	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋( + ∘ 𝐻)𝑌) = ((√‘𝑋) + (sin‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901  ⟨cop 4586   × cxp 5622  ^◡ccnv 5623   ↾ cres 5626   ∘ ccom 5628   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  1^st c1st 7931  2^nd c2nd 7932  ℂcc 11024  ℝcr 11025  0cc0 11026   + caddc 11029  +∞cpnf 11163  [,)cico 13263  √csqrt 15156  sincsin 15986

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-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

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-rmo 3350  df-reu 3351  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-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  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-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  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-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-n0 12402  df-z 12489  df-uz 12752  df-rp 12906  df-ico 13267  df-fz 13424  df-fzo 13571  df-fl 13712  df-seq 13925  df-exp 13985  df-fac 14197  df-hash 14254  df-shft 14990  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-limsup 15394  df-clim 15411  df-rlim 15412  df-sum 15610  df-ef 15990  df-sin 15992

This theorem is referenced by: (None)