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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 2728 | . . . 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 18632 | . . . 4 ⊢ (𝜑 → (( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) ↔ 0 = 0 )) |
7 | 1, 6 | mpbiri 257 | . . 3 ⊢ (𝜑 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))) |
8 | 7 | simprd 494 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) |
9 | oveq2 7434 | . . . . 5 ⊢ (𝑥 = 𝑋 → ( 0 + 𝑥) = ( 0 + 𝑋)) | |
10 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
11 | 9, 10 | eqeq12d 2744 | . . . 4 ⊢ (𝑥 = 𝑋 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 𝑋) = 𝑋)) |
12 | oveq1 7433 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 + 0 ) = (𝑋 + 0 )) | |
13 | 12, 10 | eqeq12d 2744 | . . . 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 3610 | . 2 ⊢ ((∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥) ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)) |
16 | 8, 15 | sylan 578 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ∃wrex 3067 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 0gc0g 17428 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-riota 7382 df-ov 7429 df-0g 17430 |
This theorem is referenced by: mndlrid 18720 |
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