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Theorem dvhvaddcbv 40473
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 6885 . . . . 5 (𝑓 = ℎ → (1st ‘𝑓) = (1st ‘ℎ))
32coeq1d 5855 . . . 4 (𝑓 = ℎ → ((1st ‘𝑓) ∘ (1st ‘𝑔)) = ((1st ‘ℎ) ∘ (1st ‘𝑔)))
4 fveq2 6885 . . . . 5 (𝑓 = ℎ → (2nd ‘𝑓) = (2nd ‘ℎ))
54oveq1d 7420 . . . 4 (𝑓 = ℎ → ((2nd ‘𝑓) âĻĢ (2nd ‘𝑔)) = ((2nd ‘ℎ) âĻĢ (2nd ‘𝑔)))
63, 5opeq12d 4876 . . 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 6885 . . . . 5 (𝑔 = 𝑖 → (1st ‘𝑔) = (1st ‘𝑖))
87coeq2d 5856 . . . 4 (𝑔 = 𝑖 → ((1st ‘ℎ) ∘ (1st ‘𝑔)) = ((1st ‘ℎ) ∘ (1st ‘𝑖)))
9 fveq2 6885 . . . . 5 (𝑔 = 𝑖 → (2nd ‘𝑔) = (2nd ‘𝑖))
109oveq2d 7421 . . . 4 (𝑔 = 𝑖 → ((2nd ‘ℎ) âĻĢ (2nd ‘𝑔)) = ((2nd ‘ℎ) âĻĢ (2nd ‘𝑖)))
118, 10opeq12d 4876 . . 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 7500 . 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 2754 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 1533  âŸĻcop 4629   × cxp 5667   ∘ ccom 5673  â€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  1st c1st 7972  2nd c2nd 7973
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-co 5678  df-iota 6489  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410
This theorem is referenced by:  dvhvaddval  40474
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