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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 11085 and mulcnsrec 11087. (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 11078 | . 2 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩) | |
2 | opex 5426 | . . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V | |
3 | 2 | ecid 8728 | . . 3 ⊢ [⟨𝐴, 𝐵⟩]◡ E = ⟨𝐴, 𝐵⟩ |
4 | opex 5426 | . . . 4 ⊢ ⟨𝐶, 𝐷⟩ ∈ V | |
5 | 4 | ecid 8728 | . . 3 ⊢ [⟨𝐶, 𝐷⟩]◡ E = ⟨𝐶, 𝐷⟩ |
6 | 3, 5 | oveq12i 7374 | . 2 ⊢ ([⟨𝐴, 𝐵⟩]◡ E + [⟨𝐶, 𝐷⟩]◡ E ) = (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) |
7 | opex 5426 | . . 3 ⊢ ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩ ∈ V | |
8 | 7 | ecid 8728 | . 2 ⊢ [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩]◡ E = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩ |
9 | 1, 6, 8 | 3eqtr4g 2802 | 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 397 = wceq 1542 ∈ wcel 2107 ⟨cop 4597 E cep 5541 ◡ccnv 5637 (class class class)co 7362 [cec 8653 Rcnr 10808 +R cplr 10812 + caddc 11061 |
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 5261 ax-nul 5268 ax-pr 5389 |
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-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-eprel 5542 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fv 6509 df-ov 7365 df-oprab 7366 df-ec 8657 df-c 11064 df-add 11069 |
This theorem is referenced by: axaddass 11099 axdistr 11101 |
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