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Mirrors > Home > MPE Home > Th. List > ismgmid2 | Structured version Visualization version GIF version |
Description: Show that a given element is the identity element of a magma. (Contributed by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
ismgmid.b | ⊢ 𝐵 = (Base‘𝐺) |
ismgmid.o | ⊢ 0 = (0g‘𝐺) |
ismgmid.p | ⊢ + = (+g‘𝐺) |
ismgmid2.u | ⊢ (𝜑 → 𝑈 ∈ 𝐵) |
ismgmid2.l | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈 + 𝑥) = 𝑥) |
ismgmid2.r | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑈) = 𝑥) |
Ref | Expression |
---|---|
ismgmid2 | ⊢ (𝜑 → 𝑈 = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismgmid2.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐵) | |
2 | ismgmid2.l | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈 + 𝑥) = 𝑥) | |
3 | ismgmid2.r | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑈) = 𝑥) | |
4 | 2, 3 | jca 553 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) |
5 | 4 | ralrimiva 2949 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) |
6 | ismgmid.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
7 | ismgmid.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
8 | ismgmid.p | . . . 4 ⊢ + = (+g‘𝐺) | |
9 | oveq1 6556 | . . . . . . . . 9 ⊢ (𝑒 = 𝑈 → (𝑒 + 𝑥) = (𝑈 + 𝑥)) | |
10 | 9 | eqeq1d 2612 | . . . . . . . 8 ⊢ (𝑒 = 𝑈 → ((𝑒 + 𝑥) = 𝑥 ↔ (𝑈 + 𝑥) = 𝑥)) |
11 | oveq2 6557 | . . . . . . . . 9 ⊢ (𝑒 = 𝑈 → (𝑥 + 𝑒) = (𝑥 + 𝑈)) | |
12 | 11 | eqeq1d 2612 | . . . . . . . 8 ⊢ (𝑒 = 𝑈 → ((𝑥 + 𝑒) = 𝑥 ↔ (𝑥 + 𝑈) = 𝑥)) |
13 | 10, 12 | anbi12d 743 | . . . . . . 7 ⊢ (𝑒 = 𝑈 → (((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥))) |
14 | 13 | ralbidv 2969 | . . . . . 6 ⊢ (𝑒 = 𝑈 → (∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥))) |
15 | 14 | rspcev 3282 | . . . . 5 ⊢ ((𝑈 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) |
16 | 1, 5, 15 | syl2anc 691 | . . . 4 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) |
17 | 6, 7, 8, 16 | ismgmid 17087 | . . 3 ⊢ (𝜑 → ((𝑈 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈)) |
18 | 1, 5, 17 | mpbi2and 958 | . 2 ⊢ (𝜑 → 0 = 𝑈) |
19 | 18 | eqcomd 2616 | 1 ⊢ (𝜑 → 𝑈 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-riota 6511 df-ov 6552 df-0g 15925 |
This theorem is referenced by: grpidd 17091 submnd0 17143 mnd1id 17155 frmd0 17220 mhmid 17359 cnaddid 18096 ringidss 18400 xrs10 19604 |
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