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Mirrors > Home > MPE Home > Th. List > mgmlrid | Structured version Visualization version GIF version |
Description: The identity element of a magma, if it exists, is a left and right identity. (Contributed by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
ismgmid.b | ⊢ 𝐵 = (Base‘𝐺) |
ismgmid.o | ⊢ 0 = (0g‘𝐺) |
ismgmid.p | ⊢ + = (+g‘𝐺) |
mgmidcl.e | ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) |
Ref | Expression |
---|---|
mgmlrid | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . 4 ⊢ 0 = 0 | |
2 | ismgmid.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
3 | ismgmid.o | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
4 | ismgmid.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
5 | mgmidcl.e | . . . . 5 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) | |
6 | 2, 3, 4, 5 | ismgmid 18595 | . . . 4 ⊢ (𝜑 → (( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) ↔ 0 = 0 )) |
7 | 1, 6 | mpbiri 258 | . . 3 ⊢ (𝜑 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))) |
8 | 7 | simprd 495 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) |
9 | oveq2 7412 | . . . . 5 ⊢ (𝑥 = 𝑋 → ( 0 + 𝑥) = ( 0 + 𝑋)) | |
10 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
11 | 9, 10 | eqeq12d 2742 | . . . 4 ⊢ (𝑥 = 𝑋 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 𝑋) = 𝑋)) |
12 | oveq1 7411 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 + 0 ) = (𝑋 + 0 )) | |
13 | 12, 10 | eqeq12d 2742 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 + 0 ) = 𝑥 ↔ (𝑋 + 0 ) = 𝑋)) |
14 | 11, 13 | anbi12d 630 | . . 3 ⊢ (𝑥 = 𝑋 → ((( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥) ↔ (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))) |
15 | 14 | rspccva 3605 | . 2 ⊢ ((∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥) ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)) |
16 | 8, 15 | sylan 579 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 ‘cfv 6536 (class class class)co 7404 Basecbs 17150 +gcplusg 17203 0gc0g 17391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-riota 7360 df-ov 7407 df-0g 17393 |
This theorem is referenced by: mndlrid 18683 |
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