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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 554 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) |
5 | 4 | ralrimiva 2966 | . . 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 6657 | . . . . . . . . 9 ⊢ (𝑒 = 𝑈 → (𝑒 + 𝑥) = (𝑈 + 𝑥)) | |
10 | 9 | eqeq1d 2624 | . . . . . . . 8 ⊢ (𝑒 = 𝑈 → ((𝑒 + 𝑥) = 𝑥 ↔ (𝑈 + 𝑥) = 𝑥)) |
11 | oveq2 6658 | . . . . . . . . 9 ⊢ (𝑒 = 𝑈 → (𝑥 + 𝑒) = (𝑥 + 𝑈)) | |
12 | 11 | eqeq1d 2624 | . . . . . . . 8 ⊢ (𝑒 = 𝑈 → ((𝑥 + 𝑒) = 𝑥 ↔ (𝑥 + 𝑈) = 𝑥)) |
13 | 10, 12 | anbi12d 747 | . . . . . . 7 ⊢ (𝑒 = 𝑈 → (((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥))) |
14 | 13 | ralbidv 2986 | . . . . . 6 ⊢ (𝑒 = 𝑈 → (∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥))) |
15 | 14 | rspcev 3309 | . . . . 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 693 | . . . 4 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) |
17 | 6, 7, 8, 16 | ismgmid 17264 | . . 3 ⊢ (𝜑 → ((𝑈 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈)) |
18 | 1, 5, 17 | mpbi2and 956 | . 2 ⊢ (𝜑 → 0 = 𝑈) |
19 | 18 | eqcomd 2628 | 1 ⊢ (𝜑 → 𝑈 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 0gc0g 16100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-riota 6611 df-ov 6653 df-0g 16102 |
This theorem is referenced by: grpidd 17268 submnd0 17320 mnd1id 17332 frmd0 17397 mhmid 17536 cnaddid 18273 ringidss 18577 xrs10 19785 |
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