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Theorem ex-fpar 30481
Description: Formalized example provided in the comment for fpar 8141. (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 7434 . 2 (𝑋( + ∘ 𝐻)𝑌) = (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩)
2 sqrtf 15402 . . . . . . . . 9 √:ℂ⟶ℂ
3 ffn 6736 . . . . . . . . 9 (√:ℂ⟶ℂ → √ Fn ℂ)
42, 3ax-mp 5 . . . . . . . 8 √ Fn ℂ
5 rge0ssre 13496 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ
6 ax-resscn 11212 . . . . . . . . 9 ℝ ⊆ ℂ
75, 6sstri 3993 . . . . . . . 8 (0[,)+∞) ⊆ ℂ
8 fnssres 6691 . . . . . . . . 9 ((√ Fn ℂ ∧ (0[,)+∞) ⊆ ℂ) → (√ ↾ (0[,)+∞)) Fn (0[,)+∞))
9 ex-fpar.a . . . . . . . . . . 11 𝐴 = (0[,)+∞)
109reseq2i 5994 . . . . . . . . . 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 16160 . . . . . . . . 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 5994 . . . . . . . . . 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 8141 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))
3719, 34, 36mp2an 692 . . . . 5 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
38 opex 5469 . . . . 5 ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V
3937, 38fnmpoi 8095 . . . 4 𝐻 Fn (𝐴 × 𝐵)
40 opelxpi 5722 . . . 4 ((𝑋𝐴𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
41 fvco2 7006 . . . 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 8159 . . . 4 ((𝑋𝐴𝑌𝐵) → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
4645fveq2d 6910 . . 3 ((𝑋𝐴𝑌𝐵) → ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)) = ( + ‘⟨(𝐹𝑋), (𝐺𝑌)⟩))
47 df-ov 7434 . . . 4 ((𝐹𝑋) + (𝐺𝑌)) = ( + ‘⟨(𝐹𝑋), (𝐺𝑌)⟩)
4814fveq1i 6907 . . . . . 6 (𝐹𝑋) = ((√ ↾ 𝐴)‘𝑋)
49 fvres 6925 . . . . . 6 (𝑋𝐴 → ((√ ↾ 𝐴)‘𝑋) = (√‘𝑋))
5048, 49eqtrid 2789 . . . . 5 (𝑋𝐴 → (𝐹𝑋) = (√‘𝑋))
5129fveq1i 6907 . . . . . 6 (𝐺𝑌) = ((sin ↾ 𝐵)‘𝑌)
52 fvres 6925 . . . . . 6 (𝑌𝐵 → ((sin ↾ 𝐵)‘𝑌) = (sin‘𝑌))
5351, 52eqtrid 2789 . . . . 5 (𝑌𝐵 → (𝐺𝑌) = (sin‘𝑌))
5450, 53oveqan12d 7450 . . . 4 ((𝑋𝐴𝑌𝐵) → ((𝐹𝑋) + (𝐺𝑌)) = ((√‘𝑋) + (sin‘𝑌)))
5547, 54eqtr3id 2791 . . 3 ((𝑋𝐴𝑌𝐵) → ( + ‘⟨(𝐹𝑋), (𝐺𝑌)⟩) = ((√‘𝑋) + (sin‘𝑌)))
5642, 46, 553eqtrd 2781 . 2 ((𝑋𝐴𝑌𝐵) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ((√‘𝑋) + (sin‘𝑌)))
571, 56eqtrid 2789 1 ((𝑋𝐴𝑌𝐵) → (𝑋( + ∘ 𝐻)𝑌) = ((√‘𝑋) + (sin‘𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  wss 3951  cop 4632   × cxp 5683  ccnv 5684  cres 5687  ccom 5689   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  cc 11153  cr 11154  0cc0 11155   + caddc 11158  +∞cpnf 11292  [,)cico 13389  csqrt 15272  sincsin 16099
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-ico 13393  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-fac 14313  df-hash 14370  df-shft 15106  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15507  df-clim 15524  df-rlim 15525  df-sum 15723  df-ef 16103  df-sin 16105
This theorem is referenced by: (None)
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