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Theorem dvhopaddN 41097
Description: Sum of DVecH vectors expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
dvhopadd.a 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓)𝑃(2nd𝑔))⟩)
Assertion
Ref Expression
dvhopaddN (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → (⟨𝐹, 𝑈𝐴𝐺, 𝑉⟩) = ⟨(𝐹𝐺), (𝑈𝑃𝑉)⟩)
Distinct variable groups:   𝑓,𝑔,𝐸   𝑃,𝑓,𝑔   𝑇,𝑓,𝑔
Allowed substitution hints:   𝐴(𝑓,𝑔)   𝑈(𝑓,𝑔)   𝐹(𝑓,𝑔)   𝐺(𝑓,𝑔)   𝑉(𝑓,𝑔)

Proof of Theorem dvhopaddN
StepHypRef Expression
1 opelxpi 5726 . . 3 ((𝐹𝑇𝑈𝐸) → ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸))
2 opelxpi 5726 . . 3 ((𝐺𝑇𝑉𝐸) → ⟨𝐺, 𝑉⟩ ∈ (𝑇 × 𝐸))
3 dvhopadd.a . . . 4 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓)𝑃(2nd𝑔))⟩)
43dvhvaddval 41073 . . 3 ((⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸) ∧ ⟨𝐺, 𝑉⟩ ∈ (𝑇 × 𝐸)) → (⟨𝐹, 𝑈𝐴𝐺, 𝑉⟩) = ⟨((1st ‘⟨𝐹, 𝑈⟩) ∘ (1st ‘⟨𝐺, 𝑉⟩)), ((2nd ‘⟨𝐹, 𝑈⟩)𝑃(2nd ‘⟨𝐺, 𝑉⟩))⟩)
51, 2, 4syl2an 596 . 2 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → (⟨𝐹, 𝑈𝐴𝐺, 𝑉⟩) = ⟨((1st ‘⟨𝐹, 𝑈⟩) ∘ (1st ‘⟨𝐺, 𝑉⟩)), ((2nd ‘⟨𝐹, 𝑈⟩)𝑃(2nd ‘⟨𝐺, 𝑉⟩))⟩)
6 op1stg 8025 . . . . 5 ((𝐹𝑇𝑈𝐸) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
76adantr 480 . . . 4 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
8 op1stg 8025 . . . . 5 ((𝐺𝑇𝑉𝐸) → (1st ‘⟨𝐺, 𝑉⟩) = 𝐺)
98adantl 481 . . . 4 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → (1st ‘⟨𝐺, 𝑉⟩) = 𝐺)
107, 9coeq12d 5878 . . 3 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → ((1st ‘⟨𝐹, 𝑈⟩) ∘ (1st ‘⟨𝐺, 𝑉⟩)) = (𝐹𝐺))
11 op2ndg 8026 . . . 4 ((𝐹𝑇𝑈𝐸) → (2nd ‘⟨𝐹, 𝑈⟩) = 𝑈)
12 op2ndg 8026 . . . 4 ((𝐺𝑇𝑉𝐸) → (2nd ‘⟨𝐺, 𝑉⟩) = 𝑉)
1311, 12oveqan12d 7450 . . 3 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → ((2nd ‘⟨𝐹, 𝑈⟩)𝑃(2nd ‘⟨𝐺, 𝑉⟩)) = (𝑈𝑃𝑉))
1410, 13opeq12d 4886 . 2 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → ⟨((1st ‘⟨𝐹, 𝑈⟩) ∘ (1st ‘⟨𝐺, 𝑉⟩)), ((2nd ‘⟨𝐹, 𝑈⟩)𝑃(2nd ‘⟨𝐺, 𝑉⟩))⟩ = ⟨(𝐹𝐺), (𝑈𝑃𝑉)⟩)
155, 14eqtrd 2775 1 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → (⟨𝐹, 𝑈𝐴𝐺, 𝑉⟩) = ⟨(𝐹𝐺), (𝑈𝑃𝑉)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cop 4637   × cxp 5687  ccom 5693  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012
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  ax-un 7754
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-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  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-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014
This theorem is referenced by:  dvhopN  41099
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