Theorem ex-fpar 30549

Description: Formalized example provided in the comment for fpar 8068. (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 7371	. 2 ⊢ (𝑋( + ∘ 𝐻)𝑌) = (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩)
2		sqrtf 15299	. . . . . . . . 9 ⊢ √:ℂ⟶ℂ
3		ffn 6670	. . . . . . . . 9 ⊢ (√:ℂ⟶ℂ → √ Fn ℂ)
4	2, 3	ax-mp 5	. . . . . . . 8 ⊢ √ Fn ℂ
5		rge0ssre 13384	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ
6		ax-resscn 11095	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
7	5, 6	sstri 3945	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℂ
8		fnssres 6623	. . . . . . . . 9 ⊢ ((√ Fn ℂ ∧ (0[,)+∞) ⊆ ℂ) → (√ ↾ (0[,)+∞)) Fn (0[,)+∞))
9		ex-fpar.a	. . . . . . . . . . 11 ⊢ 𝐴 = (0[,)+∞)
10	9	reseq2i 5943	. . . . . . . . . 10 ⊢ (√ ↾ 𝐴) = (√ ↾ (0[,)+∞))
11	10	fneq1i 6597	. . . . . . . . 9 ⊢ ((√ ↾ 𝐴) Fn (0[,)+∞) ↔ (√ ↾ (0[,)+∞)) Fn (0[,)+∞))
12	8, 11	sylibr 234	. . . . . . . 8 ⊢ ((√ Fn ℂ ∧ (0[,)+∞) ⊆ ℂ) → (√ ↾ 𝐴) Fn (0[,)+∞))
13	4, 7, 12	mp2an 693	. . . . . . 7 ⊢ (√ ↾ 𝐴) Fn (0[,)+∞)
14		ex-fpar.f	. . . . . . . 8 ⊢ 𝐹 = (√ ↾ 𝐴)
15		id 22	. . . . . . . . 9 ⊢ (𝐹 = (√ ↾ 𝐴) → 𝐹 = (√ ↾ 𝐴))
16	9	a1i 11	. . . . . . . . 9 ⊢ (𝐹 = (√ ↾ 𝐴) → 𝐴 = (0[,)+∞))
17	15, 16	fneq12d 6595	. . . . . . . 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 231	. . . . . 6 ⊢ 𝐹 Fn 𝐴
20		sinf 16061	. . . . . . . . 9 ⊢ sin:ℂ⟶ℂ
21		ffn 6670	. . . . . . . . 9 ⊢ (sin:ℂ⟶ℂ → sin Fn ℂ)
22	20, 21	ax-mp 5	. . . . . . . 8 ⊢ sin Fn ℂ
23		fnssres 6623	. . . . . . . . 9 ⊢ ((sin Fn ℂ ∧ ℝ ⊆ ℂ) → (sin ↾ ℝ) Fn ℝ)
24		ex-fpar.b	. . . . . . . . . . 11 ⊢ 𝐵 = ℝ
25	24	reseq2i 5943	. . . . . . . . . 10 ⊢ (sin ↾ 𝐵) = (sin ↾ ℝ)
26	25	fneq1i 6597	. . . . . . . . 9 ⊢ ((sin ↾ 𝐵) Fn ℝ ↔ (sin ↾ ℝ) Fn ℝ)
27	23, 26	sylibr 234	. . . . . . . 8 ⊢ ((sin Fn ℂ ∧ ℝ ⊆ ℂ) → (sin ↾ 𝐵) Fn ℝ)
28	22, 6, 27	mp2an 693	. . . . . . 7 ⊢ (sin ↾ 𝐵) Fn ℝ
29		ex-fpar.g	. . . . . . . 8 ⊢ 𝐺 = (sin ↾ 𝐵)
30		id 22	. . . . . . . . 9 ⊢ (𝐺 = (sin ↾ 𝐵) → 𝐺 = (sin ↾ 𝐵))
31	24	a1i 11	. . . . . . . . 9 ⊢ (𝐺 = (sin ↾ 𝐵) → 𝐵 = ℝ)
32	30, 31	fneq12d 6595	. . . . . . . 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 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)))))
36	35	fpar 8068	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩))
37	19, 34, 36	mp2an 693	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
38		opex 5419	. . . . 5 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V
39	37, 38	fnmpoi 8024	. . . 4 ⊢ 𝐻 Fn (𝐴 × 𝐵)
40		opelxpi 5669	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
41		fvco2 6939	. . . 4 ⊢ ((𝐻 Fn (𝐴 × 𝐵) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵)) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)))
42	39, 40, 41	sylancr 588	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)))
43		simpl 482	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐴)
44		simpr 484	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
45	37, 43, 44	fvproj 8086	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)
46	45	fveq2d 6846	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ( + ‘(𝐻‘⟨𝑋, 𝑌⟩)) = ( + ‘⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩))
47		df-ov 7371	. . . 4 ⊢ ((𝐹‘𝑋) + (𝐺‘𝑌)) = ( + ‘⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)
48	14	fveq1i 6843	. . . . . 6 ⊢ (𝐹‘𝑋) = ((√ ↾ 𝐴)‘𝑋)
49		fvres 6861	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → ((√ ↾ 𝐴)‘𝑋) = (√‘𝑋))
50	48, 49	eqtrid 2784	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝐹‘𝑋) = (√‘𝑋))
51	29	fveq1i 6843	. . . . . 6 ⊢ (𝐺‘𝑌) = ((sin ↾ 𝐵)‘𝑌)
52		fvres 6861	. . . . . 6 ⊢ (𝑌 ∈ 𝐵 → ((sin ↾ 𝐵)‘𝑌) = (sin‘𝑌))
53	51, 52	eqtrid 2784	. . . . 5 ⊢ (𝑌 ∈ 𝐵 → (𝐺‘𝑌) = (sin‘𝑌))
54	50, 53	oveqan12d 7387	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ((𝐹‘𝑋) + (𝐺‘𝑌)) = ((√‘𝑋) + (sin‘𝑌)))
55	47, 54	eqtr3id 2786	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ( + ‘⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩) = ((√‘𝑋) + (sin‘𝑌)))
56	42, 46, 55	3eqtrd 2776	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (( + ∘ 𝐻)‘⟨𝑋, 𝑌⟩) = ((√‘𝑋) + (sin‘𝑌)))
57	1, 56	eqtrid 2784	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋( + ∘ 𝐻)𝑌) = ((√‘𝑋) + (sin‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903  ⟨cop 4588   × cxp 5630  ^◡ccnv 5631   ↾ cres 5634   ∘ ccom 5636   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1^st c1st 7941  2^nd c2nd 7942  ℂcc 11036  ℝcr 11037  0cc0 11038   + caddc 11041  +∞cpnf 11175  [,)cico 13275  √csqrt 15168  sincsin 15998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-pm 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9357  df-inf 9358  df-oi 9427  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-n0 12414  df-z 12501  df-uz 12764  df-rp 12918  df-ico 13279  df-fz 13436  df-fzo 13583  df-fl 13724  df-seq 13937  df-exp 13997  df-fac 14209  df-hash 14266  df-shft 15002  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-limsup 15406  df-clim 15423  df-rlim 15424  df-sum 15622  df-ef 16002  df-sin 16004

This theorem is referenced by: (None)