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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 10721 and mulcnsrec 10723. (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 10714 | . 2 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (〈𝐴, 𝐵〉 + 〈𝐶, 𝐷〉) = 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉) | |
2 | opex 5333 | . . . 4 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
3 | 2 | ecid 8442 | . . 3 ⊢ [〈𝐴, 𝐵〉]◡ E = 〈𝐴, 𝐵〉 |
4 | opex 5333 | . . . 4 ⊢ 〈𝐶, 𝐷〉 ∈ V | |
5 | 4 | ecid 8442 | . . 3 ⊢ [〈𝐶, 𝐷〉]◡ E = 〈𝐶, 𝐷〉 |
6 | 3, 5 | oveq12i 7203 | . 2 ⊢ ([〈𝐴, 𝐵〉]◡ E + [〈𝐶, 𝐷〉]◡ E ) = (〈𝐴, 𝐵〉 + 〈𝐶, 𝐷〉) |
7 | opex 5333 | . . 3 ⊢ 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉 ∈ V | |
8 | 7 | ecid 8442 | . 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 399 = wceq 1543 ∈ wcel 2112 〈cop 4533 E cep 5444 ◡ccnv 5535 (class class class)co 7191 [cec 8367 Rcnr 10444 +R cplr 10448 + caddc 10697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-eprel 5445 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-oprab 7195 df-ec 8371 df-c 10700 df-add 10705 |
This theorem is referenced by: axaddass 10735 axdistr 10737 |
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