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Theorem dvhvaddcbv 39555
Description: Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.)
Hypothesis
Ref Expression
dvhvaddval.a + = (𝑓 ∈ (𝑇 × ðļ), 𝑔 ∈ (𝑇 × ðļ) â†Ķ âŸĻ((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) âĻĢ (2nd ‘𝑔))âŸĐ)
Assertion
Ref Expression
dvhvaddcbv + = (ℎ ∈ (𝑇 × ðļ), 𝑖 ∈ (𝑇 × ðļ) â†Ķ âŸĻ((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) âĻĢ (2nd ‘𝑖))âŸĐ)
Distinct variable groups:   𝑓,𝑔,ℎ,𝑖,ðļ   âĻĢ ,𝑓,𝑔,ℎ,𝑖   𝑇,𝑓,𝑔,ℎ,𝑖
Allowed substitution hints:   + (𝑓,𝑔,ℎ,𝑖)

Proof of Theorem dvhvaddcbv
StepHypRef Expression
1 dvhvaddval.a . 2 + = (𝑓 ∈ (𝑇 × ðļ), 𝑔 ∈ (𝑇 × ðļ) â†Ķ âŸĻ((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) âĻĢ (2nd ‘𝑔))âŸĐ)
2 fveq2 6843 . . . . 5 (𝑓 = ℎ → (1st ‘𝑓) = (1st ‘ℎ))
32coeq1d 5818 . . . 4 (𝑓 = ℎ → ((1st ‘𝑓) ∘ (1st ‘𝑔)) = ((1st ‘ℎ) ∘ (1st ‘𝑔)))
4 fveq2 6843 . . . . 5 (𝑓 = ℎ → (2nd ‘𝑓) = (2nd ‘ℎ))
54oveq1d 7373 . . . 4 (𝑓 = ℎ → ((2nd ‘𝑓) âĻĢ (2nd ‘𝑔)) = ((2nd ‘ℎ) âĻĢ (2nd ‘𝑔)))
63, 5opeq12d 4839 . . 3 (𝑓 = ℎ → âŸĻ((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) âĻĢ (2nd ‘𝑔))âŸĐ = âŸĻ((1st ‘ℎ) ∘ (1st ‘𝑔)), ((2nd ‘ℎ) âĻĢ (2nd ‘𝑔))âŸĐ)
7 fveq2 6843 . . . . 5 (𝑔 = 𝑖 → (1st ‘𝑔) = (1st ‘𝑖))
87coeq2d 5819 . . . 4 (𝑔 = 𝑖 → ((1st ‘ℎ) ∘ (1st ‘𝑔)) = ((1st ‘ℎ) ∘ (1st ‘𝑖)))
9 fveq2 6843 . . . . 5 (𝑔 = 𝑖 → (2nd ‘𝑔) = (2nd ‘𝑖))
109oveq2d 7374 . . . 4 (𝑔 = 𝑖 → ((2nd ‘ℎ) âĻĢ (2nd ‘𝑔)) = ((2nd ‘ℎ) âĻĢ (2nd ‘𝑖)))
118, 10opeq12d 4839 . . 3 (𝑔 = 𝑖 → âŸĻ((1st ‘ℎ) ∘ (1st ‘𝑔)), ((2nd ‘ℎ) âĻĢ (2nd ‘𝑔))âŸĐ = âŸĻ((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) âĻĢ (2nd ‘𝑖))âŸĐ)
126, 11cbvmpov 7453 . 2 (𝑓 ∈ (𝑇 × ðļ), 𝑔 ∈ (𝑇 × ðļ) â†Ķ âŸĻ((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) âĻĢ (2nd ‘𝑔))âŸĐ) = (ℎ ∈ (𝑇 × ðļ), 𝑖 ∈ (𝑇 × ðļ) â†Ķ âŸĻ((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) âĻĢ (2nd ‘𝑖))âŸĐ)
131, 12eqtri 2765 1 + = (ℎ ∈ (𝑇 × ðļ), 𝑖 ∈ (𝑇 × ðļ) â†Ķ âŸĻ((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) âĻĢ (2nd ‘𝑖))âŸĐ)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  âŸĻcop 4593   × cxp 5632   ∘ ccom 5638  â€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1st c1st 7920  2nd c2nd 7921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-co 5643  df-iota 6449  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363
This theorem is referenced by:  dvhvaddval  39556
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