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Theorem dvhopaddN 40643
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 5709 . . 3 ((𝐹𝑇𝑈𝐸) → ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸))
2 opelxpi 5709 . . 3 ((𝐺𝑇𝑉𝐸) → ⟨𝐺, 𝑉⟩ ∈ (𝑇 × 𝐸))
3 dvhopadd.a . . . 4 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓)𝑃(2nd𝑔))⟩)
43dvhvaddval 40619 . . 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 594 . 2 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → (⟨𝐹, 𝑈𝐴𝐺, 𝑉⟩) = ⟨((1st ‘⟨𝐹, 𝑈⟩) ∘ (1st ‘⟨𝐺, 𝑉⟩)), ((2nd ‘⟨𝐹, 𝑈⟩)𝑃(2nd ‘⟨𝐺, 𝑉⟩))⟩)
6 op1stg 8003 . . . . 5 ((𝐹𝑇𝑈𝐸) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
76adantr 479 . . . 4 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
8 op1stg 8003 . . . . 5 ((𝐺𝑇𝑉𝐸) → (1st ‘⟨𝐺, 𝑉⟩) = 𝐺)
98adantl 480 . . . 4 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → (1st ‘⟨𝐺, 𝑉⟩) = 𝐺)
107, 9coeq12d 5861 . . 3 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → ((1st ‘⟨𝐹, 𝑈⟩) ∘ (1st ‘⟨𝐺, 𝑉⟩)) = (𝐹𝐺))
11 op2ndg 8004 . . . 4 ((𝐹𝑇𝑈𝐸) → (2nd ‘⟨𝐹, 𝑈⟩) = 𝑈)
12 op2ndg 8004 . . . 4 ((𝐺𝑇𝑉𝐸) → (2nd ‘⟨𝐺, 𝑉⟩) = 𝑉)
1311, 12oveqan12d 7435 . . 3 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → ((2nd ‘⟨𝐹, 𝑈⟩)𝑃(2nd ‘⟨𝐺, 𝑉⟩)) = (𝑈𝑃𝑉))
1410, 13opeq12d 4877 . 2 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → ⟨((1st ‘⟨𝐹, 𝑈⟩) ∘ (1st ‘⟨𝐺, 𝑉⟩)), ((2nd ‘⟨𝐹, 𝑈⟩)𝑃(2nd ‘⟨𝐺, 𝑉⟩))⟩ = ⟨(𝐹𝐺), (𝑈𝑃𝑉)⟩)
155, 14eqtrd 2765 1 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → (⟨𝐹, 𝑈𝐴𝐺, 𝑉⟩) = ⟨(𝐹𝐺), (𝑈𝑃𝑉)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cop 4630   × cxp 5670  ccom 5676  cfv 6543  (class class class)co 7416  cmpo 7418  1st c1st 7989  2nd c2nd 7990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992
This theorem is referenced by:  dvhopN  40645
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