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Theorem ex-fpar 28399
Description: Formalized example provided in the comment for fpar 7837. (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 7173 . 2 (𝑋( + ∘ 𝐻)𝑌) = (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩)
2 sqrtf 14813 . . . . . . . . 9 √:ℂ⟶ℂ
3 ffn 6504 . . . . . . . . 9 (√:ℂ⟶ℂ → √ Fn ℂ)
42, 3ax-mp 5 . . . . . . . 8 √ Fn ℂ
5 rge0ssre 12930 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ
6 ax-resscn 10672 . . . . . . . . 9 ℝ ⊆ ℂ
75, 6sstri 3886 . . . . . . . 8 (0[,)+∞) ⊆ ℂ
8 fnssres 6459 . . . . . . . . 9 ((√ Fn ℂ ∧ (0[,)+∞) ⊆ ℂ) → (√ ↾ (0[,)+∞)) Fn (0[,)+∞))
9 ex-fpar.a . . . . . . . . . . 11 𝐴 = (0[,)+∞)
109reseq2i 5822 . . . . . . . . . 10 (√ ↾ 𝐴) = (√ ↾ (0[,)+∞))
1110fneq1i 6435 . . . . . . . . 9 ((√ ↾ 𝐴) Fn (0[,)+∞) ↔ (√ ↾ (0[,)+∞)) Fn (0[,)+∞))
128, 11sylibr 237 . . . . . . . 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 6433 . . . . . . . 8 (𝐹 = (√ ↾ 𝐴) → (𝐹 Fn 𝐴 ↔ (√ ↾ 𝐴) Fn (0[,)+∞)))
1814, 17ax-mp 5 . . . . . . 7 (𝐹 Fn 𝐴 ↔ (√ ↾ 𝐴) Fn (0[,)+∞))
1913, 18mpbir 234 . . . . . 6 𝐹 Fn 𝐴
20 sinf 15569 . . . . . . . . 9 sin:ℂ⟶ℂ
21 ffn 6504 . . . . . . . . 9 (sin:ℂ⟶ℂ → sin Fn ℂ)
2220, 21ax-mp 5 . . . . . . . 8 sin Fn ℂ
23 fnssres 6459 . . . . . . . . 9 ((sin Fn ℂ ∧ ℝ ⊆ ℂ) → (sin ↾ ℝ) Fn ℝ)
24 ex-fpar.b . . . . . . . . . . 11 𝐵 = ℝ
2524reseq2i 5822 . . . . . . . . . 10 (sin ↾ 𝐵) = (sin ↾ ℝ)
2625fneq1i 6435 . . . . . . . . 9 ((sin ↾ 𝐵) Fn ℝ ↔ (sin ↾ ℝ) Fn ℝ)
2723, 26sylibr 237 . . . . . . . 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 6433 . . . . . . . 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 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)))))
3635fpar 7837 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))
3719, 34, 36mp2an 692 . . . . 5 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
38 opex 5322 . . . . 5 ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V
3937, 38fnmpoi 7793 . . . 4 𝐻 Fn (𝐴 × 𝐵)
40 opelxpi 5562 . . . 4 ((𝑋𝐴𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
41 fvco2 6765 . . . 4 ((𝐻 Fn (𝐴 × 𝐵) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵)) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)))
4239, 40, 41sylancr 590 . . 3 ((𝑋𝐴𝑌𝐵) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)))
43 simpl 486 . . . . 5 ((𝑋𝐴𝑌𝐵) → 𝑋𝐴)
44 simpr 488 . . . . 5 ((𝑋𝐴𝑌𝐵) → 𝑌𝐵)
4537, 43, 44fvproj 7854 . . . 4 ((𝑋𝐴𝑌𝐵) → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
4645fveq2d 6678 . . 3 ((𝑋𝐴𝑌𝐵) → ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)) = ( + ‘⟨(𝐹𝑋), (𝐺𝑌)⟩))
47 df-ov 7173 . . . 4 ((𝐹𝑋) + (𝐺𝑌)) = ( + ‘⟨(𝐹𝑋), (𝐺𝑌)⟩)
4814fveq1i 6675 . . . . . 6 (𝐹𝑋) = ((√ ↾ 𝐴)‘𝑋)
49 fvres 6693 . . . . . 6 (𝑋𝐴 → ((√ ↾ 𝐴)‘𝑋) = (√‘𝑋))
5048, 49syl5eq 2785 . . . . 5 (𝑋𝐴 → (𝐹𝑋) = (√‘𝑋))
5129fveq1i 6675 . . . . . 6 (𝐺𝑌) = ((sin ↾ 𝐵)‘𝑌)
52 fvres 6693 . . . . . 6 (𝑌𝐵 → ((sin ↾ 𝐵)‘𝑌) = (sin‘𝑌))
5351, 52syl5eq 2785 . . . . 5 (𝑌𝐵 → (𝐺𝑌) = (sin‘𝑌))
5450, 53oveqan12d 7189 . . . 4 ((𝑋𝐴𝑌𝐵) → ((𝐹𝑋) + (𝐺𝑌)) = ((√‘𝑋) + (sin‘𝑌)))
5547, 54eqtr3id 2787 . . 3 ((𝑋𝐴𝑌𝐵) → ( + ‘⟨(𝐹𝑋), (𝐺𝑌)⟩) = ((√‘𝑋) + (sin‘𝑌)))
5642, 46, 553eqtrd 2777 . 2 ((𝑋𝐴𝑌𝐵) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ((√‘𝑋) + (sin‘𝑌)))
571, 56syl5eq 2785 1 ((𝑋𝐴𝑌𝐵) → (𝑋( + ∘ 𝐻)𝑌) = ((√‘𝑋) + (sin‘𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  Vcvv 3398  cin 3842  wss 3843  cop 4522   × cxp 5523  ccnv 5524  cres 5527  ccom 5529   Fn wfn 6334  wf 6335  cfv 6339  (class class class)co 7170  cmpo 7172  1st c1st 7712  2nd c2nd 7713  cc 10613  cr 10614  0cc0 10615   + caddc 10618  +∞cpnf 10750  [,)cico 12823  csqrt 14682  sincsin 15509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-inf2 9177  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-pre-sup 10693
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-pm 8440  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-sup 8979  df-inf 8980  df-oi 9047  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-2 11779  df-3 11780  df-n0 11977  df-z 12063  df-uz 12325  df-rp 12473  df-ico 12827  df-fz 12982  df-fzo 13125  df-fl 13253  df-seq 13461  df-exp 13522  df-fac 13726  df-hash 13783  df-shft 14516  df-cj 14548  df-re 14549  df-im 14550  df-sqrt 14684  df-abs 14685  df-limsup 14918  df-clim 14935  df-rlim 14936  df-sum 15136  df-ef 15513  df-sin 15515
This theorem is referenced by: (None)
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