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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 11182 and mulcnsrec 11184. (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 11175 | . 2 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (〈𝐴, 𝐵〉 + 〈𝐶, 𝐷〉) = 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉) | |
| 2 | opex 5469 | . . . 4 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
| 3 | 2 | ecid 8822 | . . 3 ⊢ [〈𝐴, 𝐵〉]◡ E = 〈𝐴, 𝐵〉 | 
| 4 | opex 5469 | . . . 4 ⊢ 〈𝐶, 𝐷〉 ∈ V | |
| 5 | 4 | ecid 8822 | . . 3 ⊢ [〈𝐶, 𝐷〉]◡ E = 〈𝐶, 𝐷〉 | 
| 6 | 3, 5 | oveq12i 7443 | . 2 ⊢ ([〈𝐴, 𝐵〉]◡ E + [〈𝐶, 𝐷〉]◡ E ) = (〈𝐴, 𝐵〉 + 〈𝐶, 𝐷〉) | 
| 7 | opex 5469 | . . 3 ⊢ 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉 ∈ V | |
| 8 | 7 | ecid 8822 | . 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 395 = wceq 1540 ∈ wcel 2108 〈cop 4632 E cep 5583 ◡ccnv 5684 (class class class)co 7431 [cec 8743 Rcnr 10905 +R cplr 10909 + caddc 11158 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-ec 8747 df-c 11161 df-add 11166 | 
| This theorem is referenced by: axaddass 11196 axdistr 11198 | 
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