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Theorem imasaddvallem 16170
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 6638 . 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 16169 . . . . 5 (𝜑 Fn (𝐵 × 𝐵))
6 fnfun 5976 . . . . 5 ( Fn (𝐵 × 𝐵) → Fun )
75, 6syl 17 . . . 4 (𝜑 → Fun )
873ad2ant1 1080 . . 3 ((𝜑𝑋𝑉𝑌𝑉) → Fun )
9 fveq2 6178 . . . . . . . . . . 11 (𝑝 = 𝑋 → (𝐹𝑝) = (𝐹𝑋))
109opeq1d 4399 . . . . . . . . . 10 (𝑝 = 𝑋 → ⟨(𝐹𝑝), (𝐹𝑌)⟩ = ⟨(𝐹𝑋), (𝐹𝑌)⟩)
11 oveq1 6642 . . . . . . . . . . 11 (𝑝 = 𝑋 → (𝑝 · 𝑌) = (𝑋 · 𝑌))
1211fveq2d 6182 . . . . . . . . . 10 (𝑝 = 𝑋 → (𝐹‘(𝑝 · 𝑌)) = (𝐹‘(𝑋 · 𝑌)))
1310, 12opeq12d 4401 . . . . . . . . 9 (𝑝 = 𝑋 → ⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩ = ⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩)
1413sneqd 4180 . . . . . . . 8 (𝑝 = 𝑋 → {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} = {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩})
1514ssiun2s 4555 . . . . . . 7 (𝑋𝑉 → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
16153ad2ant2 1081 . . . . . 6 ((𝜑𝑋𝑉𝑌𝑉) → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
17 fveq2 6178 . . . . . . . . . . . . 13 (𝑞 = 𝑌 → (𝐹𝑞) = (𝐹𝑌))
1817opeq2d 4400 . . . . . . . . . . . 12 (𝑞 = 𝑌 → ⟨(𝐹𝑝), (𝐹𝑞)⟩ = ⟨(𝐹𝑝), (𝐹𝑌)⟩)
19 oveq2 6643 . . . . . . . . . . . . 13 (𝑞 = 𝑌 → (𝑝 · 𝑞) = (𝑝 · 𝑌))
2019fveq2d 6182 . . . . . . . . . . . 12 (𝑞 = 𝑌 → (𝐹‘(𝑝 · 𝑞)) = (𝐹‘(𝑝 · 𝑌)))
2118, 20opeq12d 4401 . . . . . . . . . . 11 (𝑞 = 𝑌 → ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ = ⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩)
2221sneqd 4180 . . . . . . . . . 10 (𝑞 = 𝑌 → {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} = {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
2322ssiun2s 4555 . . . . . . . . 9 (𝑌𝑉 → {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2423ralrimivw 2964 . . . . . . . 8 (𝑌𝑉 → ∀𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
25 ss2iun 4527 . . . . . . . 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 1082 . . . . . 6 ((𝜑𝑋𝑉𝑌𝑉) → 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2816, 27sstrd 3605 . . . . 5 ((𝜑𝑋𝑉𝑌𝑉) → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2943ad2ant1 1080 . . . . 5 ((𝜑𝑋𝑉𝑌𝑉) → = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
3028, 29sseqtr4d 3634 . . . 4 ((𝜑𝑋𝑉𝑌𝑉) → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ )
31 opex 4923 . . . . 5 ⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ V
3231snss 4307 . . . 4 (⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ↔ {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ )
3330, 32sylibr 224 . . 3 ((𝜑𝑋𝑉𝑌𝑉) → ⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ )
34 funopfv 6222 . . 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 2666 1 ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌)) = (𝐹‘(𝑋 · 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  wss 3567  {csn 4168  cop 4174   ciun 4511   × cxp 5102  Fun wfun 5870   Fn wfn 5871  ontowfo 5874  cfv 5876  (class class class)co 6635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fo 5882  df-fv 5884  df-ov 6638
This theorem is referenced by:  imasaddval  16173  imasmulval  16176  qusaddvallem  16192
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