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Mirrors > Home > MPE Home > Th. List > addcnsrec | Structured version Visualization version GIF version |
Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 11165 and mulcnsrec 11167. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
addcnsrec | ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → ([⟨𝐴, 𝐵⟩]◡ E + [⟨𝐶, 𝐷⟩]◡ E ) = [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩]◡ E ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcnsr 11158 | . 2 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩) | |
2 | opex 5460 | . . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V | |
3 | 2 | ecid 8799 | . . 3 ⊢ [⟨𝐴, 𝐵⟩]◡ E = ⟨𝐴, 𝐵⟩ |
4 | opex 5460 | . . . 4 ⊢ ⟨𝐶, 𝐷⟩ ∈ V | |
5 | 4 | ecid 8799 | . . 3 ⊢ [⟨𝐶, 𝐷⟩]◡ E = ⟨𝐶, 𝐷⟩ |
6 | 3, 5 | oveq12i 7428 | . 2 ⊢ ([⟨𝐴, 𝐵⟩]◡ E + [⟨𝐶, 𝐷⟩]◡ E ) = (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) |
7 | opex 5460 | . . 3 ⊢ ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩ ∈ V | |
8 | 7 | ecid 8799 | . 2 ⊢ [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩]◡ E = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩ |
9 | 1, 6, 8 | 3eqtr4g 2790 | 1 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → ([⟨𝐴, 𝐵⟩]◡ E + [⟨𝐶, 𝐷⟩]◡ E ) = [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩]◡ E ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⟨cop 4630 E cep 5575 ◡ccnv 5671 (class class class)co 7416 [cec 8721 Rcnr 10888 +R cplr 10892 + caddc 11141 |
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 |
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-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 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-id 5570 df-eprel 5576 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7419 df-oprab 7420 df-ec 8725 df-c 11144 df-add 11149 |
This theorem is referenced by: axaddass 11179 axdistr 11181 |
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