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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 512 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) |
5 | 4 | ralrimiva 3179 | . . 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 7152 | . . . . . . . 8 ⊢ (𝑒 = 𝑈 → (𝑒 + 𝑥) = (𝑈 + 𝑥)) | |
10 | 9 | eqeq1d 2820 | . . . . . . 7 ⊢ (𝑒 = 𝑈 → ((𝑒 + 𝑥) = 𝑥 ↔ (𝑈 + 𝑥) = 𝑥)) |
11 | 10 | ovanraleqv 7169 | . . . . . 6 ⊢ (𝑒 = 𝑈 → (∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥))) |
12 | 11 | rspcev 3620 | . . . . 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 584 | . . . 4 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) |
14 | 6, 7, 8, 13 | ismgmid 17863 | . . 3 ⊢ (𝜑 → ((𝑈 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈)) |
15 | 1, 5, 14 | mpbi2and 708 | . 2 ⊢ (𝜑 → 0 = 𝑈) |
16 | 15 | eqcomd 2824 | 1 ⊢ (𝜑 → 𝑈 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 0gc0g 16701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-riota 7103 df-ov 7148 df-0g 16703 |
This theorem is referenced by: lidrididd 17868 grpidd 17869 submnd0 17928 mnd1id 17941 frmd0 18013 pwmndid 18039 mhmid 18158 cnaddid 18919 ringidss 19256 xrs10 20512 efmndid 43985 |
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