Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  imasaddvallem Structured version   Visualization version   GIF version

 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.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)

StepHypRef Expression
1 df-ov 6552 . 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 16011 . . . . 5 (𝜑 Fn (𝐵 × 𝐵))
6 fnfun 5902 . . . . 5 ( Fn (𝐵 × 𝐵) → Fun )
75, 6syl 17 . . . 4 (𝜑 → Fun )
873ad2ant1 1075 . . 3 ((𝜑𝑋𝑉𝑌𝑉) → Fun )
9 fveq2 6103 . . . . . . . . . . 11 (𝑝 = 𝑋 → (𝐹𝑝) = (𝐹𝑋))
109opeq1d 4346 . . . . . . . . . 10 (𝑝 = 𝑋 → ⟨(𝐹𝑝), (𝐹𝑌)⟩ = ⟨(𝐹𝑋), (𝐹𝑌)⟩)
11 oveq1 6556 . . . . . . . . . . 11 (𝑝 = 𝑋 → (𝑝 · 𝑌) = (𝑋 · 𝑌))
1211fveq2d 6107 . . . . . . . . . 10 (𝑝 = 𝑋 → (𝐹‘(𝑝 · 𝑌)) = (𝐹‘(𝑋 · 𝑌)))
1310, 12opeq12d 4348 . . . . . . . . 9 (𝑝 = 𝑋 → ⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩ = ⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩)
1413sneqd 4137 . . . . . . . 8 (𝑝 = 𝑋 → {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} = {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩})
1514ssiun2s 4500 . . . . . . 7 (𝑋𝑉 → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
16153ad2ant2 1076 . . . . . 6 ((𝜑𝑋𝑉𝑌𝑉) → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
17 fveq2 6103 . . . . . . . . . . . . 13 (𝑞 = 𝑌 → (𝐹𝑞) = (𝐹𝑌))
1817opeq2d 4347 . . . . . . . . . . . 12 (𝑞 = 𝑌 → ⟨(𝐹𝑝), (𝐹𝑞)⟩ = ⟨(𝐹𝑝), (𝐹𝑌)⟩)
19 oveq2 6557 . . . . . . . . . . . . 13 (𝑞 = 𝑌 → (𝑝 · 𝑞) = (𝑝 · 𝑌))
2019fveq2d 6107 . . . . . . . . . . . 12 (𝑞 = 𝑌 → (𝐹‘(𝑝 · 𝑞)) = (𝐹‘(𝑝 · 𝑌)))
2118, 20opeq12d 4348 . . . . . . . . . . 11 (𝑞 = 𝑌 → ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ = ⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩)
2221sneqd 4137 . . . . . . . . . 10 (𝑞 = 𝑌 → {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} = {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
2322ssiun2s 4500 . . . . . . . . 9 (𝑌𝑉 → {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2423ralrimivw 2950 . . . . . . . 8 (𝑌𝑉 → ∀𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
25 ss2iun 4472 . . . . . . . 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 1077 . . . . . 6 ((𝜑𝑋𝑉𝑌𝑉) → 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2816, 27sstrd 3578 . . . . 5 ((𝜑𝑋𝑉𝑌𝑉) → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2943ad2ant1 1075 . . . . 5 ((𝜑𝑋𝑉𝑌𝑉) → = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
3028, 29sseqtr4d 3605 . . . 4 ((𝜑𝑋𝑉𝑌𝑉) → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ )
31 opex 4859 . . . . 5 ⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ V
3231snss 4259 . . . 4 (⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ↔ {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ )
3330, 32sylibr 223 . . 3 ((𝜑𝑋𝑉𝑌𝑉) → ⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ )
34 funopfv 6145 . . 3 (Fun → (⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ → ( ‘⟨(𝐹𝑋), (𝐹𝑌)⟩) = (𝐹‘(𝑋 · 𝑌))))
358, 33, 34sylc 63 . 2 ((𝜑𝑋𝑉𝑌𝑉) → ( ‘⟨(𝐹𝑋), (𝐹𝑌)⟩) = (𝐹‘(𝑋 · 𝑌)))
361, 35syl5eq 2656 1 ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌)) = (𝐹‘(𝑋 · 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  {csn 4125  ⟨cop 4131  ∪ ciun 4455   × cxp 5036  Fun wfun 5798   Fn wfn 5799  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-ov 6552 This theorem is referenced by:  imasaddval  16015  imasmulval  16018  qusaddvallem  16034
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