![]() |
Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
|
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 37330 | . . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) |
4 | 3 | eqcomd 2734 | . . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) = 𝑈) |
5 | 1, 2 | iorlid 37331 | . . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 ∈ 𝑋) |
6 | 1 | exidu1 37329 | . . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) |
7 | oveq1 7427 | . . . . . . 7 ⊢ (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥)) | |
8 | 7 | eqeq1d 2730 | . . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥)) |
9 | 8 | ovanraleqv 7444 | . . . . 5 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))) |
10 | 9 | riota2 7402 | . . . 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 583 | . . 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 7428 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑈𝐺𝑥) = (𝑈𝐺𝐴)) | |
14 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
15 | 13, 14 | eqeq12d 2744 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑈𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝐴) = 𝐴)) |
16 | oveq1 7427 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑈) = (𝐴𝐺𝑈)) | |
17 | 16, 14 | eqeq12d 2744 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑈) = 𝑥 ↔ (𝐴𝐺𝑈) = 𝐴)) |
18 | 15, 17 | anbi12d 631 | . . 3 ⊢ (𝑥 = 𝐴 → (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ↔ ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴))) |
19 | 18 | rspccva 3608 | . 2 ⊢ ((∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴)) |
20 | 12, 19 | sylan 579 | 1 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ∃!wreu 3371 ∩ cin 3946 ran crn 5679 ‘cfv 6548 ℩crio 7375 (class class class)co 7420 GIdcgi 30313 ExId cexid 37317 Magmacmagm 37321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fo 6554 df-fv 6556 df-riota 7376 df-ov 7423 df-gid 30317 df-exid 37318 df-mgmOLD 37322 |
This theorem is referenced by: exidreslem 37350 rngoidmlem 37409 |
Copyright terms: Public domain | W3C validator |