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Theorem dvhopaddN 36925
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 5289 . . 3 ((𝐹𝑇𝑈𝐸) → ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸))
2 opelxpi 5289 . . 3 ((𝐺𝑇𝑉𝐸) → ⟨𝐺, 𝑉⟩ ∈ (𝑇 × 𝐸))
3 dvhopadd.a . . . 4 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓)𝑃(2nd𝑔))⟩)
43dvhvaddval 36901 . . 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 577 . 2 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → (⟨𝐹, 𝑈𝐴𝐺, 𝑉⟩) = ⟨((1st ‘⟨𝐹, 𝑈⟩) ∘ (1st ‘⟨𝐺, 𝑉⟩)), ((2nd ‘⟨𝐹, 𝑈⟩)𝑃(2nd ‘⟨𝐺, 𝑉⟩))⟩)
6 op1stg 7328 . . . . 5 ((𝐹𝑇𝑈𝐸) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
76adantr 466 . . . 4 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
8 op1stg 7328 . . . . 5 ((𝐺𝑇𝑉𝐸) → (1st ‘⟨𝐺, 𝑉⟩) = 𝐺)
98adantl 467 . . . 4 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → (1st ‘⟨𝐺, 𝑉⟩) = 𝐺)
107, 9coeq12d 5426 . . 3 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → ((1st ‘⟨𝐹, 𝑈⟩) ∘ (1st ‘⟨𝐺, 𝑉⟩)) = (𝐹𝐺))
11 op2ndg 7329 . . . 4 ((𝐹𝑇𝑈𝐸) → (2nd ‘⟨𝐹, 𝑈⟩) = 𝑈)
12 op2ndg 7329 . . . 4 ((𝐺𝑇𝑉𝐸) → (2nd ‘⟨𝐺, 𝑉⟩) = 𝑉)
1311, 12oveqan12d 6813 . . 3 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → ((2nd ‘⟨𝐹, 𝑈⟩)𝑃(2nd ‘⟨𝐺, 𝑉⟩)) = (𝑈𝑃𝑉))
1410, 13opeq12d 4548 . 2 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → ⟨((1st ‘⟨𝐹, 𝑈⟩) ∘ (1st ‘⟨𝐺, 𝑉⟩)), ((2nd ‘⟨𝐹, 𝑈⟩)𝑃(2nd ‘⟨𝐺, 𝑉⟩))⟩ = ⟨(𝐹𝐺), (𝑈𝑃𝑉)⟩)
155, 14eqtrd 2805 1 (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → (⟨𝐹, 𝑈𝐴𝐺, 𝑉⟩) = ⟨(𝐹𝐺), (𝑈𝑃𝑉)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cop 4323   × cxp 5248  ccom 5254  cfv 6032  (class class class)co 6794  cmpt2 6796  1st c1st 7314  2nd c2nd 7315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5995  df-fun 6034  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-1st 7316  df-2nd 7317
This theorem is referenced by:  dvhopN  36927
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