Theorem ex-fpar 30842

Description: Formalized example provided in the comment for fpar 8113. (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 7419	. 2 ⊢ (𝑋( + ∘ 𝐻)𝑌) = (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩)
2		sqrtf 15434	. . . . . . . . 9 ⊢ √:ℂ⟶ℂ
3		ffn 6709	. . . . . . . . 9 ⊢ (√:ℂ⟶ℂ → √ Fn ℂ)
4	2, 3	ax-mp 5	. . . . . . . 8 ⊢ √ Fn ℂ
5		rge0ssre 13494	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ
6		ax-resscn 11168	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
7	5, 6	sstri 3947	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℂ
8		fnssres 6662	. . . . . . . . 9 ⊢ ((√ Fn ℂ ∧ (0[,)+∞) ⊆ ℂ) → (√ ↾ (0[,)+∞)) Fn (0[,)+∞))
9		ex-fpar.a	. . . . . . . . . . 11 ⊢ 𝐴 = (0[,)+∞)
10	9	reseq2i 5977	. . . . . . . . . 10 ⊢ (√ ↾ 𝐴) = (√ ↾ (0[,)+∞))
11	10	fneq1i 6636	. . . . . . . . 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 705	. . . . . . 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 6634	. . . . . . . 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 16197	. . . . . . . . 9 ⊢ sin:ℂ⟶ℂ
21		ffn 6709	. . . . . . . . 9 ⊢ (sin:ℂ⟶ℂ → sin Fn ℂ)
22	20, 21	ax-mp 5	. . . . . . . 8 ⊢ sin Fn ℂ
23		fnssres 6662	. . . . . . . . 9 ⊢ ((sin Fn ℂ ∧ ℝ ⊆ ℂ) → (sin ↾ ℝ) Fn ℝ)
24		ex-fpar.b	. . . . . . . . . . 11 ⊢ 𝐵 = ℝ
25	24	reseq2i 5977	. . . . . . . . . 10 ⊢ (sin ↾ 𝐵) = (sin ↾ ℝ)
26	25	fneq1i 6636	. . . . . . . . 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 705	. . . . . . 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 6634	. . . . . . . 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 8113	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩))
37	19, 34, 36	mp2an 705	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
38		opex 5447	. . . . 5 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V
39	37, 38	fnmpoi 8069	. . . 4 ⊢ 𝐻 Fn (𝐴 × 𝐵)
40		opelxpi 5700	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
41		fvco2 6982	. . . 4 ⊢ ((𝐻 Fn (𝐴 × 𝐵) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵)) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)))
42	39, 40, 41	sylancr 599	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)))
43		simpl 488	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐴)
44		simpr 490	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
45	37, 43, 44	fvproj 8132	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)
46	45	fveq2d 6889	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)) = ( + ‘⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩))
47		df-ov 7419	. . . 4 ⊢ ((𝐹‘𝑋) + (𝐺‘𝑌)) = ( + ‘⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)
48	14	fveq1i 6886	. . . . . 6 ⊢ (𝐹‘𝑋) = ((√ ↾ 𝐴)‘𝑋)
49		fvres 6904	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → ((√ ↾ 𝐴)‘𝑋) = (√‘𝑋))
50	48, 49	eqtrid 2812	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝐹‘𝑋) = (√‘𝑋))
51	29	fveq1i 6886	. . . . . 6 ⊢ (𝐺‘𝑌) = ((sin ↾ 𝐵)‘𝑌)
52		fvres 6904	. . . . . 6 ⊢ (𝑌 ∈ 𝐵 → ((sin ↾ 𝐵)‘𝑌) = (sin‘𝑌))
53	51, 52	eqtrid 2812	. . . . 5 ⊢ (𝑌 ∈ 𝐵 → (𝐺‘𝑌) = (sin‘𝑌))
54	50, 53	oveqan12d 7435	. . . 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ∩ cin 3905   ⊆ wss 3906  ⟨cop 4597   × cxp 5661  ^◡ccnv 5662   ↾ cres 5665   ∘ ccom 5667   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416   ∈ cmpo 7418  1^st c1st 7986  2^nd c2nd 7987  ℂcc 11109  ℝcr 11110  0cc0 11111   + caddc 11114  +∞cpnf 11251  [,)cico 13385  √csqrt 15303  sincsin 16134

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-inf2 9613  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188  ax-pre-sup 11189

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-pm 8829  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9405  df-inf 9406  df-oi 9475  df-card 9937  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-div 11883  df-nn 12245  df-2 12314  df-3 12315  df-n0 12516  df-z 12603  df-uz 12874  df-rp 13028  df-ico 13389  df-fz 13547  df-fzo 13695  df-fl 13838  df-seq 14051  df-exp 14111  df-fac 14323  df-hash 14380  df-shft 15123  df-cj 15169  df-re 15170  df-im 15171  df-sqrt 15305  df-abs 15306  df-limsup 15541  df-clim 15558  df-rlim 15559  df-sum 15757  df-ef 16138  df-sin 16140

This theorem is used by: (None)