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Theorem ex-fpar 28826
Description: Formalized example provided in the comment for fpar 7956. (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 ((𝑋𝐴𝑌𝐵) → (𝑋( + ∘ 𝐻)𝑌) = ((√‘𝑋) + (sin‘𝑌)))

Proof of Theorem ex-fpar
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 7278 . 2 (𝑋( + ∘ 𝐻)𝑌) = (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩)
2 sqrtf 15075 . . . . . . . . 9 √:ℂ⟶ℂ
3 ffn 6600 . . . . . . . . 9 (√:ℂ⟶ℂ → √ Fn ℂ)
42, 3ax-mp 5 . . . . . . . 8 √ Fn ℂ
5 rge0ssre 13188 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ
6 ax-resscn 10928 . . . . . . . . 9 ℝ ⊆ ℂ
75, 6sstri 3930 . . . . . . . 8 (0[,)+∞) ⊆ ℂ
8 fnssres 6555 . . . . . . . . 9 ((√ Fn ℂ ∧ (0[,)+∞) ⊆ ℂ) → (√ ↾ (0[,)+∞)) Fn (0[,)+∞))
9 ex-fpar.a . . . . . . . . . . 11 𝐴 = (0[,)+∞)
109reseq2i 5888 . . . . . . . . . 10 (√ ↾ 𝐴) = (√ ↾ (0[,)+∞))
1110fneq1i 6530 . . . . . . . . 9 ((√ ↾ 𝐴) Fn (0[,)+∞) ↔ (√ ↾ (0[,)+∞)) Fn (0[,)+∞))
128, 11sylibr 233 . . . . . . . 8 ((√ Fn ℂ ∧ (0[,)+∞) ⊆ ℂ) → (√ ↾ 𝐴) Fn (0[,)+∞))
134, 7, 12mp2an 689 . . . . . . 7 (√ ↾ 𝐴) Fn (0[,)+∞)
14 ex-fpar.f . . . . . . . 8 𝐹 = (√ ↾ 𝐴)
15 id 22 . . . . . . . . 9 (𝐹 = (√ ↾ 𝐴) → 𝐹 = (√ ↾ 𝐴))
169a1i 11 . . . . . . . . 9 (𝐹 = (√ ↾ 𝐴) → 𝐴 = (0[,)+∞))
1715, 16fneq12d 6528 . . . . . . . 8 (𝐹 = (√ ↾ 𝐴) → (𝐹 Fn 𝐴 ↔ (√ ↾ 𝐴) Fn (0[,)+∞)))
1814, 17ax-mp 5 . . . . . . 7 (𝐹 Fn 𝐴 ↔ (√ ↾ 𝐴) Fn (0[,)+∞))
1913, 18mpbir 230 . . . . . 6 𝐹 Fn 𝐴
20 sinf 15833 . . . . . . . . 9 sin:ℂ⟶ℂ
21 ffn 6600 . . . . . . . . 9 (sin:ℂ⟶ℂ → sin Fn ℂ)
2220, 21ax-mp 5 . . . . . . . 8 sin Fn ℂ
23 fnssres 6555 . . . . . . . . 9 ((sin Fn ℂ ∧ ℝ ⊆ ℂ) → (sin ↾ ℝ) Fn ℝ)
24 ex-fpar.b . . . . . . . . . . 11 𝐵 = ℝ
2524reseq2i 5888 . . . . . . . . . 10 (sin ↾ 𝐵) = (sin ↾ ℝ)
2625fneq1i 6530 . . . . . . . . 9 ((sin ↾ 𝐵) Fn ℝ ↔ (sin ↾ ℝ) Fn ℝ)
2723, 26sylibr 233 . . . . . . . 8 ((sin Fn ℂ ∧ ℝ ⊆ ℂ) → (sin ↾ 𝐵) Fn ℝ)
2822, 6, 27mp2an 689 . . . . . . 7 (sin ↾ 𝐵) Fn ℝ
29 ex-fpar.g . . . . . . . 8 𝐺 = (sin ↾ 𝐵)
30 id 22 . . . . . . . . 9 (𝐺 = (sin ↾ 𝐵) → 𝐺 = (sin ↾ 𝐵))
3124a1i 11 . . . . . . . . 9 (𝐺 = (sin ↾ 𝐵) → 𝐵 = ℝ)
3230, 31fneq12d 6528 . . . . . . . 8 (𝐺 = (sin ↾ 𝐵) → (𝐺 Fn 𝐵 ↔ (sin ↾ 𝐵) Fn ℝ))
3329, 32ax-mp 5 . . . . . . 7 (𝐺 Fn 𝐵 ↔ (sin ↾ 𝐵) Fn ℝ)
3428, 33mpbir 230 . . . . . 6 𝐺 Fn 𝐵
35 ex-fpar.h . . . . . . 7 𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))
3635fpar 7956 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))
3719, 34, 36mp2an 689 . . . . 5 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
38 opex 5379 . . . . 5 ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V
3937, 38fnmpoi 7910 . . . 4 𝐻 Fn (𝐴 × 𝐵)
40 opelxpi 5626 . . . 4 ((𝑋𝐴𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
41 fvco2 6865 . . . 4 ((𝐻 Fn (𝐴 × 𝐵) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵)) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)))
4239, 40, 41sylancr 587 . . 3 ((𝑋𝐴𝑌𝐵) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)))
43 simpl 483 . . . . 5 ((𝑋𝐴𝑌𝐵) → 𝑋𝐴)
44 simpr 485 . . . . 5 ((𝑋𝐴𝑌𝐵) → 𝑌𝐵)
4537, 43, 44fvproj 7975 . . . 4 ((𝑋𝐴𝑌𝐵) → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
4645fveq2d 6778 . . 3 ((𝑋𝐴𝑌𝐵) → ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)) = ( + ‘⟨(𝐹𝑋), (𝐺𝑌)⟩))
47 df-ov 7278 . . . 4 ((𝐹𝑋) + (𝐺𝑌)) = ( + ‘⟨(𝐹𝑋), (𝐺𝑌)⟩)
4814fveq1i 6775 . . . . . 6 (𝐹𝑋) = ((√ ↾ 𝐴)‘𝑋)
49 fvres 6793 . . . . . 6 (𝑋𝐴 → ((√ ↾ 𝐴)‘𝑋) = (√‘𝑋))
5048, 49eqtrid 2790 . . . . 5 (𝑋𝐴 → (𝐹𝑋) = (√‘𝑋))
5129fveq1i 6775 . . . . . 6 (𝐺𝑌) = ((sin ↾ 𝐵)‘𝑌)
52 fvres 6793 . . . . . 6 (𝑌𝐵 → ((sin ↾ 𝐵)‘𝑌) = (sin‘𝑌))
5351, 52eqtrid 2790 . . . . 5 (𝑌𝐵 → (𝐺𝑌) = (sin‘𝑌))
5450, 53oveqan12d 7294 . . . 4 ((𝑋𝐴𝑌𝐵) → ((𝐹𝑋) + (𝐺𝑌)) = ((√‘𝑋) + (sin‘𝑌)))
5547, 54eqtr3id 2792 . . 3 ((𝑋𝐴𝑌𝐵) → ( + ‘⟨(𝐹𝑋), (𝐺𝑌)⟩) = ((√‘𝑋) + (sin‘𝑌)))
5642, 46, 553eqtrd 2782 . 2 ((𝑋𝐴𝑌𝐵) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ((√‘𝑋) + (sin‘𝑌)))
571, 56eqtrid 2790 1 ((𝑋𝐴𝑌𝐵) → (𝑋( + ∘ 𝐻)𝑌) = ((√‘𝑋) + (sin‘𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  Vcvv 3432  cin 3886  wss 3887  cop 4567   × cxp 5587  ccnv 5588  cres 5591  ccom 5593   Fn wfn 6428  wf 6429  cfv 6433  (class class class)co 7275  cmpo 7277  1st c1st 7829  2nd c2nd 7830  cc 10869  cr 10870  0cc0 10871   + caddc 10874  +∞cpnf 11006  [,)cico 13081  csqrt 14944  sincsin 15773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-ico 13085  df-fz 13240  df-fzo 13383  df-fl 13512  df-seq 13722  df-exp 13783  df-fac 13988  df-hash 14045  df-shft 14778  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-limsup 15180  df-clim 15197  df-rlim 15198  df-sum 15398  df-ef 15777  df-sin 15779
This theorem is referenced by: (None)
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