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Theorem imasaddvallem 16236
Description: The operation of an image structure is defined to distribute over the mapping function. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
imasaddf.f (𝜑𝐹:𝑉onto𝐵)
imasaddf.e ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
imasaddflem.a (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
Assertion
Ref Expression
imasaddvallem ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌)) = (𝐹‘(𝑋 · 𝑌)))
Distinct variable groups:   𝑞,𝑝,𝐵   𝑎,𝑏,𝑝,𝑞,𝑉   · ,𝑝,𝑞   𝑋,𝑝   𝐹,𝑎,𝑏,𝑝,𝑞   𝜑,𝑎,𝑏,𝑝,𝑞   ,𝑎,𝑏,𝑝,𝑞   𝑌,𝑝,𝑞
Allowed substitution hints:   𝐵(𝑎,𝑏)   · (𝑎,𝑏)   𝑋(𝑞,𝑎,𝑏)   𝑌(𝑎,𝑏)

Proof of Theorem imasaddvallem
StepHypRef Expression
1 df-ov 6693 . 2 ((𝐹𝑋) (𝐹𝑌)) = ( ‘⟨(𝐹𝑋), (𝐹𝑌)⟩)
2 imasaddf.f . . . . . 6 (𝜑𝐹:𝑉onto𝐵)
3 imasaddf.e . . . . . 6 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
4 imasaddflem.a . . . . . 6 (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
52, 3, 4imasaddfnlem 16235 . . . . 5 (𝜑 Fn (𝐵 × 𝐵))
6 fnfun 6026 . . . . 5 ( Fn (𝐵 × 𝐵) → Fun )
75, 6syl 17 . . . 4 (𝜑 → Fun )
873ad2ant1 1102 . . 3 ((𝜑𝑋𝑉𝑌𝑉) → Fun )
9 fveq2 6229 . . . . . . . . . . 11 (𝑝 = 𝑋 → (𝐹𝑝) = (𝐹𝑋))
109opeq1d 4439 . . . . . . . . . 10 (𝑝 = 𝑋 → ⟨(𝐹𝑝), (𝐹𝑌)⟩ = ⟨(𝐹𝑋), (𝐹𝑌)⟩)
11 oveq1 6697 . . . . . . . . . . 11 (𝑝 = 𝑋 → (𝑝 · 𝑌) = (𝑋 · 𝑌))
1211fveq2d 6233 . . . . . . . . . 10 (𝑝 = 𝑋 → (𝐹‘(𝑝 · 𝑌)) = (𝐹‘(𝑋 · 𝑌)))
1310, 12opeq12d 4441 . . . . . . . . 9 (𝑝 = 𝑋 → ⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩ = ⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩)
1413sneqd 4222 . . . . . . . 8 (𝑝 = 𝑋 → {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} = {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩})
1514ssiun2s 4596 . . . . . . 7 (𝑋𝑉 → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
16153ad2ant2 1103 . . . . . 6 ((𝜑𝑋𝑉𝑌𝑉) → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
17 fveq2 6229 . . . . . . . . . . . . 13 (𝑞 = 𝑌 → (𝐹𝑞) = (𝐹𝑌))
1817opeq2d 4440 . . . . . . . . . . . 12 (𝑞 = 𝑌 → ⟨(𝐹𝑝), (𝐹𝑞)⟩ = ⟨(𝐹𝑝), (𝐹𝑌)⟩)
19 oveq2 6698 . . . . . . . . . . . . 13 (𝑞 = 𝑌 → (𝑝 · 𝑞) = (𝑝 · 𝑌))
2019fveq2d 6233 . . . . . . . . . . . 12 (𝑞 = 𝑌 → (𝐹‘(𝑝 · 𝑞)) = (𝐹‘(𝑝 · 𝑌)))
2118, 20opeq12d 4441 . . . . . . . . . . 11 (𝑞 = 𝑌 → ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ = ⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩)
2221sneqd 4222 . . . . . . . . . 10 (𝑞 = 𝑌 → {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} = {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
2322ssiun2s 4596 . . . . . . . . 9 (𝑌𝑉 → {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2423ralrimivw 2996 . . . . . . . 8 (𝑌𝑉 → ∀𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
25 ss2iun 4568 . . . . . . . 8 (∀𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} → 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2624, 25syl 17 . . . . . . 7 (𝑌𝑉 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
27263ad2ant3 1104 . . . . . 6 ((𝜑𝑋𝑉𝑌𝑉) → 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2816, 27sstrd 3646 . . . . 5 ((𝜑𝑋𝑉𝑌𝑉) → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2943ad2ant1 1102 . . . . 5 ((𝜑𝑋𝑉𝑌𝑉) → = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
3028, 29sseqtr4d 3675 . . . 4 ((𝜑𝑋𝑉𝑌𝑉) → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ )
31 opex 4962 . . . . 5 ⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ V
3231snss 4348 . . . 4 (⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ↔ {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ )
3330, 32sylibr 224 . . 3 ((𝜑𝑋𝑉𝑌𝑉) → ⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ )
34 funopfv 6273 . . 3 (Fun → (⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ → ( ‘⟨(𝐹𝑋), (𝐹𝑌)⟩) = (𝐹‘(𝑋 · 𝑌))))
358, 33, 34sylc 65 . 2 ((𝜑𝑋𝑉𝑌𝑉) → ( ‘⟨(𝐹𝑋), (𝐹𝑌)⟩) = (𝐹‘(𝑋 · 𝑌)))
361, 35syl5eq 2697 1 ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌)) = (𝐹‘(𝑋 · 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wss 3607  {csn 4210  cop 4216   ciun 4552   × cxp 5141  Fun wfun 5920   Fn wfn 5921  ontowfo 5924  cfv 5926  (class class class)co 6690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-ov 6693
This theorem is referenced by:  imasaddval  16239  imasmulval  16242  qusaddvallem  16258
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