![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11119 and mulcnsrec 11121. (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 11112 | . 2 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (〈𝐴, 𝐵〉 + 〈𝐶, 𝐷〉) = 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉) | |
2 | opex 5457 | . . . 4 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
3 | 2 | ecid 8759 | . . 3 ⊢ [〈𝐴, 𝐵〉]◡ E = 〈𝐴, 𝐵〉 |
4 | opex 5457 | . . . 4 ⊢ 〈𝐶, 𝐷〉 ∈ V | |
5 | 4 | ecid 8759 | . . 3 ⊢ [〈𝐶, 𝐷〉]◡ E = 〈𝐶, 𝐷〉 |
6 | 3, 5 | oveq12i 7405 | . 2 ⊢ ([〈𝐴, 𝐵〉]◡ E + [〈𝐶, 𝐷〉]◡ E ) = (〈𝐴, 𝐵〉 + 〈𝐶, 𝐷〉) |
7 | opex 5457 | . . 3 ⊢ 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉 ∈ V | |
8 | 7 | ecid 8759 | . 2 ⊢ [〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉]◡ E = 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉 |
9 | 1, 6, 8 | 3eqtr4g 2796 | 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 396 = wceq 1541 ∈ wcel 2106 〈cop 4628 E cep 5572 ◡ccnv 5668 (class class class)co 7393 [cec 8684 Rcnr 10842 +R cplr 10846 + caddc 11095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-id 5567 df-eprel 5573 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fv 6540 df-ov 7396 df-oprab 7397 df-ec 8688 df-c 11098 df-add 11103 |
This theorem is referenced by: axaddass 11133 axdistr 11135 |
Copyright terms: Public domain | W3C validator |