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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 10564 and mulcnsrec 10566. (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 10557 | . 2 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (〈𝐴, 𝐵〉 + 〈𝐶, 𝐷〉) = 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉) | |
2 | opex 5356 | . . . 4 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
3 | 2 | ecid 8362 | . . 3 ⊢ [〈𝐴, 𝐵〉]◡ E = 〈𝐴, 𝐵〉 |
4 | opex 5356 | . . . 4 ⊢ 〈𝐶, 𝐷〉 ∈ V | |
5 | 4 | ecid 8362 | . . 3 ⊢ [〈𝐶, 𝐷〉]◡ E = 〈𝐶, 𝐷〉 |
6 | 3, 5 | oveq12i 7168 | . 2 ⊢ ([〈𝐴, 𝐵〉]◡ E + [〈𝐶, 𝐷〉]◡ E ) = (〈𝐴, 𝐵〉 + 〈𝐶, 𝐷〉) |
7 | opex 5356 | . . 3 ⊢ 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉 ∈ V | |
8 | 7 | ecid 8362 | . 2 ⊢ [〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉]◡ E = 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉 |
9 | 1, 6, 8 | 3eqtr4g 2881 | 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 398 = wceq 1537 ∈ wcel 2114 〈cop 4573 E cep 5464 ◡ccnv 5554 (class class class)co 7156 [cec 8287 Rcnr 10287 +R cplr 10291 + caddc 10540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-eprel 5465 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-ec 8291 df-c 10543 df-add 10548 |
This theorem is referenced by: axaddass 10578 axdistr 10580 |
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