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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 515 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) |
5 | 4 | ralrimiva 3105 | . . 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 7220 | . . . . . . . 8 ⊢ (𝑒 = 𝑈 → (𝑒 + 𝑥) = (𝑈 + 𝑥)) | |
10 | 9 | eqeq1d 2739 | . . . . . . 7 ⊢ (𝑒 = 𝑈 → ((𝑒 + 𝑥) = 𝑥 ↔ (𝑈 + 𝑥) = 𝑥)) |
11 | 10 | ovanraleqv 7237 | . . . . . 6 ⊢ (𝑒 = 𝑈 → (∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥))) |
12 | 11 | rspcev 3537 | . . . . 5 ⊢ ((𝑈 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) |
13 | 1, 5, 12 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) |
14 | 6, 7, 8, 13 | ismgmid 18137 | . . 3 ⊢ (𝜑 → ((𝑈 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈)) |
15 | 1, 5, 14 | mpbi2and 712 | . 2 ⊢ (𝜑 → 0 = 𝑈) |
16 | 15 | eqcomd 2743 | 1 ⊢ (𝜑 → 𝑈 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∃wrex 3062 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 +gcplusg 16802 0gc0g 16944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-riota 7170 df-ov 7216 df-0g 16946 |
This theorem is referenced by: lidrididd 18142 grpidd 18143 submnd0 18202 mnd1id 18215 frmd0 18287 efmndid 18315 pwmndid 18363 mhmid 18484 cnaddid 19255 ringidss 19595 xrs10 20402 idlsrg0g 31365 |
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