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Mirrors > Home > MPE Home > Th. List > Mathboxes > cmpidelt | Structured version Visualization version GIF version |
Description: A magma right and left identity element keeps the other elements unchanged. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cmpidelt.1 | ⊢ 𝑋 = ran 𝐺 |
cmpidelt.2 | ⊢ 𝑈 = (GId‘𝐺) |
Ref | Expression |
---|---|
cmpidelt | ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmpidelt.1 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
2 | cmpidelt.2 | . . . . 5 ⊢ 𝑈 = (GId‘𝐺) | |
3 | 1, 2 | idrval 37844 | . . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) |
4 | 3 | eqcomd 2741 | . . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) = 𝑈) |
5 | 1, 2 | iorlid 37845 | . . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 ∈ 𝑋) |
6 | 1 | exidu1 37843 | . . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) |
7 | oveq1 7438 | . . . . . . 7 ⊢ (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥)) | |
8 | 7 | eqeq1d 2737 | . . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥)) |
9 | 8 | ovanraleqv 7455 | . . . . 5 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))) |
10 | 9 | riota2 7413 | . . . 4 ⊢ ((𝑈 ∈ 𝑋 ∧ ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → (∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ↔ (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) = 𝑈)) |
11 | 5, 6, 10 | syl2anc 584 | . . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ↔ (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) = 𝑈)) |
12 | 4, 11 | mpbird 257 | . 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)) |
13 | oveq2 7439 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑈𝐺𝑥) = (𝑈𝐺𝐴)) | |
14 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
15 | 13, 14 | eqeq12d 2751 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑈𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝐴) = 𝐴)) |
16 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑈) = (𝐴𝐺𝑈)) | |
17 | 16, 14 | eqeq12d 2751 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑈) = 𝑥 ↔ (𝐴𝐺𝑈) = 𝐴)) |
18 | 15, 17 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝐴 → (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ↔ ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴))) |
19 | 18 | rspccva 3621 | . 2 ⊢ ((∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴)) |
20 | 12, 19 | sylan 580 | 1 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃!wreu 3376 ∩ cin 3962 ran crn 5690 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 GIdcgi 30519 ExId cexid 37831 Magmacmagm 37835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-riota 7388 df-ov 7434 df-gid 30523 df-exid 37832 df-mgmOLD 37836 |
This theorem is referenced by: exidreslem 37864 rngoidmlem 37923 |
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