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Theorem ex-fpar 30490
Description: Formalized example provided in the comment for fpar 8139. (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 7433 . 2 (𝑋( + ∘ 𝐻)𝑌) = (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩)
2 sqrtf 15398 . . . . . . . . 9 √:ℂ⟶ℂ
3 ffn 6736 . . . . . . . . 9 (√:ℂ⟶ℂ → √ Fn ℂ)
42, 3ax-mp 5 . . . . . . . 8 √ Fn ℂ
5 rge0ssre 13492 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ
6 ax-resscn 11209 . . . . . . . . 9 ℝ ⊆ ℂ
75, 6sstri 4004 . . . . . . . 8 (0[,)+∞) ⊆ ℂ
8 fnssres 6691 . . . . . . . . 9 ((√ Fn ℂ ∧ (0[,)+∞) ⊆ ℂ) → (√ ↾ (0[,)+∞)) Fn (0[,)+∞))
9 ex-fpar.a . . . . . . . . . . 11 𝐴 = (0[,)+∞)
109reseq2i 5996 . . . . . . . . . 10 (√ ↾ 𝐴) = (√ ↾ (0[,)+∞))
1110fneq1i 6665 . . . . . . . . 9 ((√ ↾ 𝐴) Fn (0[,)+∞) ↔ (√ ↾ (0[,)+∞)) Fn (0[,)+∞))
128, 11sylibr 234 . . . . . . . 8 ((√ Fn ℂ ∧ (0[,)+∞) ⊆ ℂ) → (√ ↾ 𝐴) Fn (0[,)+∞))
134, 7, 12mp2an 692 . . . . . . 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 6663 . . . . . . . 8 (𝐹 = (√ ↾ 𝐴) → (𝐹 Fn 𝐴 ↔ (√ ↾ 𝐴) Fn (0[,)+∞)))
1814, 17ax-mp 5 . . . . . . 7 (𝐹 Fn 𝐴 ↔ (√ ↾ 𝐴) Fn (0[,)+∞))
1913, 18mpbir 231 . . . . . 6 𝐹 Fn 𝐴
20 sinf 16156 . . . . . . . . 9 sin:ℂ⟶ℂ
21 ffn 6736 . . . . . . . . 9 (sin:ℂ⟶ℂ → sin Fn ℂ)
2220, 21ax-mp 5 . . . . . . . 8 sin Fn ℂ
23 fnssres 6691 . . . . . . . . 9 ((sin Fn ℂ ∧ ℝ ⊆ ℂ) → (sin ↾ ℝ) Fn ℝ)
24 ex-fpar.b . . . . . . . . . . 11 𝐵 = ℝ
2524reseq2i 5996 . . . . . . . . . 10 (sin ↾ 𝐵) = (sin ↾ ℝ)
2625fneq1i 6665 . . . . . . . . 9 ((sin ↾ 𝐵) Fn ℝ ↔ (sin ↾ ℝ) Fn ℝ)
2723, 26sylibr 234 . . . . . . . 8 ((sin Fn ℂ ∧ ℝ ⊆ ℂ) → (sin ↾ 𝐵) Fn ℝ)
2822, 6, 27mp2an 692 . . . . . . 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 6663 . . . . . . . 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 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)))))
3635fpar 8139 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))
3719, 34, 36mp2an 692 . . . . 5 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
38 opex 5474 . . . . 5 ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V
3937, 38fnmpoi 8093 . . . 4 𝐻 Fn (𝐴 × 𝐵)
40 opelxpi 5725 . . . 4 ((𝑋𝐴𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
41 fvco2 7005 . . . 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 482 . . . . 5 ((𝑋𝐴𝑌𝐵) → 𝑋𝐴)
44 simpr 484 . . . . 5 ((𝑋𝐴𝑌𝐵) → 𝑌𝐵)
4537, 43, 44fvproj 8157 . . . 4 ((𝑋𝐴𝑌𝐵) → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
4645fveq2d 6910 . . 3 ((𝑋𝐴𝑌𝐵) → ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)) = ( + ‘⟨(𝐹𝑋), (𝐺𝑌)⟩))
47 df-ov 7433 . . . 4 ((𝐹𝑋) + (𝐺𝑌)) = ( + ‘⟨(𝐹𝑋), (𝐺𝑌)⟩)
4814fveq1i 6907 . . . . . 6 (𝐹𝑋) = ((√ ↾ 𝐴)‘𝑋)
49 fvres 6925 . . . . . 6 (𝑋𝐴 → ((√ ↾ 𝐴)‘𝑋) = (√‘𝑋))
5048, 49eqtrid 2786 . . . . 5 (𝑋𝐴 → (𝐹𝑋) = (√‘𝑋))
5129fveq1i 6907 . . . . . 6 (𝐺𝑌) = ((sin ↾ 𝐵)‘𝑌)
52 fvres 6925 . . . . . 6 (𝑌𝐵 → ((sin ↾ 𝐵)‘𝑌) = (sin‘𝑌))
5351, 52eqtrid 2786 . . . . 5 (𝑌𝐵 → (𝐺𝑌) = (sin‘𝑌))
5450, 53oveqan12d 7449 . . . 4 ((𝑋𝐴𝑌𝐵) → ((𝐹𝑋) + (𝐺𝑌)) = ((√‘𝑋) + (sin‘𝑌)))
5547, 54eqtr3id 2788 . . 3 ((𝑋𝐴𝑌𝐵) → ( + ‘⟨(𝐹𝑋), (𝐺𝑌)⟩) = ((√‘𝑋) + (sin‘𝑌)))
5642, 46, 553eqtrd 2778 . 2 ((𝑋𝐴𝑌𝐵) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ((√‘𝑋) + (sin‘𝑌)))
571, 56eqtrid 2786 1 ((𝑋𝐴𝑌𝐵) → (𝑋( + ∘ 𝐻)𝑌) = ((√‘𝑋) + (sin‘𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  Vcvv 3477  cin 3961  wss 3962  cop 4636   × cxp 5686  ccnv 5687  cres 5690  ccom 5692   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  cmpo 7432  1st c1st 8010  2nd c2nd 8011  cc 11150  cr 11151  0cc0 11152   + caddc 11155  +∞cpnf 11289  [,)cico 13385  csqrt 15268  sincsin 16095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-ico 13389  df-fz 13544  df-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-fac 14309  df-hash 14366  df-shft 15102  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-limsup 15503  df-clim 15520  df-rlim 15521  df-sum 15719  df-ef 16099  df-sin 16101
This theorem is referenced by: (None)
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