| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > mulcnsrec | Structured version Visualization version GIF version | ||
| Description: Technical trick to permit
re-use of some equivalence class lemmas for
operation laws. The trick involves ecid 8822,
which shows that the coset of
the converse membership relation (which is not an equivalence relation)
leaves a set unchanged. See also dfcnqs 11182.
Note: This is the last lemma (from which the axioms will be derived) in the construction of real and complex numbers. The construction starts at cnpi 10884. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| mulcnsrec | ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → ([〈𝐴, 𝐵〉]◡ E · [〈𝐶, 𝐷〉]◡ E ) = [〈((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))〉]◡ E ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mulcnsr 11176 | . 2 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (〈𝐴, 𝐵〉 · 〈𝐶, 𝐷〉) = 〈((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·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) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))〉 ∈ V | |
| 8 | 7 | ecid 8822 | . 2 ⊢ [〈((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))〉]◡ E = 〈((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·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) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·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 -1Rcm1r 10908 +R cplr 10909 ·R cmr 10910 · cmul 11160 |
| 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-mul 11167 |
| This theorem is referenced by: axmulcom 11195 axmulass 11197 axdistr 11198 |
| Copyright terms: Public domain | W3C validator |