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Mirrors > Home > MPE Home > Th. List > mgmidcl | Structured version Visualization version GIF version |
Description: The identity element of a magma, if it exists, belongs to the base set. (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 |
---|---|
mgmidcl | ⊢ (𝜑 → 0 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2730 | . . 3 ⊢ 0 = 0 | |
2 | ismgmid.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | ismgmid.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
4 | ismgmid.p | . . . 4 ⊢ + = (+g‘𝐺) | |
5 | mgmidcl.e | . . . 4 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) | |
6 | 2, 3, 4, 5 | ismgmid 18590 | . . 3 ⊢ (𝜑 → (( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) ↔ 0 = 0 )) |
7 | 1, 6 | mpbiri 257 | . 2 ⊢ (𝜑 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))) |
8 | 7 | simpld 493 | 1 ⊢ (𝜑 → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 ‘cfv 6542 (class class class)co 7411 Basecbs 17148 +gcplusg 17201 0gc0g 17389 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-riota 7367 df-ov 7414 df-0g 17391 |
This theorem is referenced by: mndidcl 18674 |
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