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Mirrors > Home > MPE Home > Th. List > mndpfo | Structured version Visualization version GIF version |
Description: The addition operation of a monoid as a function is an onto function. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 11-Oct-2013.) (Revised by AV, 23-Jan-2020.) |
Ref | Expression |
---|---|
mndpf.b | ⊢ 𝐵 = (Base‘𝐺) |
mndpf.p | ⊢ ⨣ = (+𝑓‘𝐺) |
Ref | Expression |
---|---|
mndpfo | ⊢ (𝐺 ∈ Mnd → ⨣ :(𝐵 × 𝐵)–onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndpf.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | mndpf.p | . . 3 ⊢ ⨣ = (+𝑓‘𝐺) | |
3 | 1, 2 | mndplusf 18712 | . 2 ⊢ (𝐺 ∈ Mnd → ⨣ :(𝐵 × 𝐵)⟶𝐵) |
4 | simpr 484 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
5 | eqid 2728 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
6 | 1, 5 | mndidcl 18709 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵) |
7 | 6 | adantr 480 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (0g‘𝐺) ∈ 𝐵) |
8 | eqid 2728 | . . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
9 | 1, 8, 5 | mndrid 18715 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥) |
10 | 9 | eqcomd 2734 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → 𝑥 = (𝑥(+g‘𝐺)(0g‘𝐺))) |
11 | rspceov 7467 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ (0g‘𝐺) ∈ 𝐵 ∧ 𝑥 = (𝑥(+g‘𝐺)(0g‘𝐺))) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦(+g‘𝐺)𝑧)) | |
12 | 4, 7, 10, 11 | syl3anc 1369 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦(+g‘𝐺)𝑧)) |
13 | 1, 8, 2 | plusfval 18607 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦 ⨣ 𝑧) = (𝑦(+g‘𝐺)𝑧)) |
14 | 13 | eqeq2d 2739 | . . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥 = (𝑦 ⨣ 𝑧) ↔ 𝑥 = (𝑦(+g‘𝐺)𝑧))) |
15 | 14 | 2rexbiia 3212 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦(+g‘𝐺)𝑧)) |
16 | 12, 15 | sylibr 233 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧)) |
17 | 16 | ralrimiva 3143 | . 2 ⊢ (𝐺 ∈ Mnd → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧)) |
18 | foov 7595 | . 2 ⊢ ( ⨣ :(𝐵 × 𝐵)–onto→𝐵 ↔ ( ⨣ :(𝐵 × 𝐵)⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧))) | |
19 | 3, 17, 18 | sylanbrc 582 | 1 ⊢ (𝐺 ∈ Mnd → ⨣ :(𝐵 × 𝐵)–onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ∃wrex 3067 × cxp 5676 ⟶wf 6544 –onto→wfo 6546 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 +gcplusg 17233 0gc0g 17421 +𝑓cplusf 18597 Mndcmnd 18694 |
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-pow 5365 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-pw 4605 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-res 5690 df-ima 5691 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-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-0g 17423 df-plusf 18599 df-mgm 18600 df-sgrp 18679 df-mnd 18695 |
This theorem is referenced by: mndfo 18718 grpplusfo 18906 |
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