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Theorem imasaddvallem 17576
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 7434 . 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 17575 . . . . 5 (𝜑 Fn (𝐵 × 𝐵))
6 fnfun 6669 . . . . 5 ( Fn (𝐵 × 𝐵) → Fun )
75, 6syl 17 . . . 4 (𝜑 → Fun )
873ad2ant1 1132 . . 3 ((𝜑𝑋𝑉𝑌𝑉) → Fun )
9 fveq2 6907 . . . . . . . . . . 11 (𝑝 = 𝑋 → (𝐹𝑝) = (𝐹𝑋))
109opeq1d 4884 . . . . . . . . . 10 (𝑝 = 𝑋 → ⟨(𝐹𝑝), (𝐹𝑌)⟩ = ⟨(𝐹𝑋), (𝐹𝑌)⟩)
11 fvoveq1 7454 . . . . . . . . . 10 (𝑝 = 𝑋 → (𝐹‘(𝑝 · 𝑌)) = (𝐹‘(𝑋 · 𝑌)))
1210, 11opeq12d 4886 . . . . . . . . 9 (𝑝 = 𝑋 → ⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩ = ⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩)
1312sneqd 4643 . . . . . . . 8 (𝑝 = 𝑋 → {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} = {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩})
1413ssiun2s 5053 . . . . . . 7 (𝑋𝑉 → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
15143ad2ant2 1133 . . . . . 6 ((𝜑𝑋𝑉𝑌𝑉) → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
16 fveq2 6907 . . . . . . . . . . . . 13 (𝑞 = 𝑌 → (𝐹𝑞) = (𝐹𝑌))
1716opeq2d 4885 . . . . . . . . . . . 12 (𝑞 = 𝑌 → ⟨(𝐹𝑝), (𝐹𝑞)⟩ = ⟨(𝐹𝑝), (𝐹𝑌)⟩)
18 oveq2 7439 . . . . . . . . . . . . 13 (𝑞 = 𝑌 → (𝑝 · 𝑞) = (𝑝 · 𝑌))
1918fveq2d 6911 . . . . . . . . . . . 12 (𝑞 = 𝑌 → (𝐹‘(𝑝 · 𝑞)) = (𝐹‘(𝑝 · 𝑌)))
2017, 19opeq12d 4886 . . . . . . . . . . 11 (𝑞 = 𝑌 → ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ = ⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩)
2120sneqd 4643 . . . . . . . . . 10 (𝑞 = 𝑌 → {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} = {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
2221ssiun2s 5053 . . . . . . . . 9 (𝑌𝑉 → {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2322ralrimivw 3148 . . . . . . . 8 (𝑌𝑉 → ∀𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
24 ss2iun 5015 . . . . . . . 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))⟩})
2523, 24syl 17 . . . . . . 7 (𝑌𝑉 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
26253ad2ant3 1134 . . . . . 6 ((𝜑𝑋𝑉𝑌𝑉) → 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2715, 26sstrd 4006 . . . . 5 ((𝜑𝑋𝑉𝑌𝑉) → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2843ad2ant1 1132 . . . . 5 ((𝜑𝑋𝑉𝑌𝑉) → = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2927, 28sseqtrrd 4037 . . . 4 ((𝜑𝑋𝑉𝑌𝑉) → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ )
30 opex 5475 . . . . 5 ⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ V
3130snss 4790 . . . 4 (⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ↔ {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ )
3229, 31sylibr 234 . . 3 ((𝜑𝑋𝑉𝑌𝑉) → ⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ )
33 funopfv 6959 . . 3 (Fun → (⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ → ( ‘⟨(𝐹𝑋), (𝐹𝑌)⟩) = (𝐹‘(𝑋 · 𝑌))))
348, 32, 33sylc 65 . 2 ((𝜑𝑋𝑉𝑌𝑉) → ( ‘⟨(𝐹𝑋), (𝐹𝑌)⟩) = (𝐹‘(𝑋 · 𝑌)))
351, 34eqtrid 2787 1 ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌)) = (𝐹‘(𝑋 · 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wss 3963  {csn 4631  cop 4637   ciun 4996   × cxp 5687  Fun wfun 6557   Fn wfn 6558  ontowfo 6561  cfv 6563  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-ov 7434
This theorem is referenced by:  imasaddval  17579  imasmulval  17582  qusaddvallem  17598
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