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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 8776,
which shows that the coset of
the converse membership relation (which is not an equivalence relation)
leaves a set unchanged. See also dfcnqs 11137.
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 10839. (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 11131 | . 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 5465 | . . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V | |
3 | 2 | ecid 8776 | . . 3 ⊢ [⟨𝐴, 𝐵⟩]◡ E = ⟨𝐴, 𝐵⟩ |
4 | opex 5465 | . . . 4 ⊢ ⟨𝐶, 𝐷⟩ ∈ V | |
5 | 4 | ecid 8776 | . . 3 ⊢ [⟨𝐶, 𝐷⟩]◡ E = ⟨𝐶, 𝐷⟩ |
6 | 3, 5 | oveq12i 7421 | . 2 ⊢ ([⟨𝐴, 𝐵⟩]◡ E · [⟨𝐶, 𝐷⟩]◡ E ) = (⟨𝐴, 𝐵⟩ · ⟨𝐶, 𝐷⟩) |
7 | opex 5465 | . . 3 ⊢ ⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩ ∈ V | |
8 | 7 | ecid 8776 | . 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 2798 | 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 397 = wceq 1542 ∈ wcel 2107 ⟨cop 4635 E cep 5580 ◡ccnv 5676 (class class class)co 7409 [cec 8701 Rcnr 10860 -1Rcm1r 10863 +R cplr 10864 ·R cmr 10865 · cmul 11115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-eprel 5581 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-oprab 7413 df-ec 8705 df-c 11116 df-mul 11122 |
This theorem is referenced by: axmulcom 11150 axmulass 11152 axdistr 11153 |
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