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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 17921 | . 2 ⊢ (𝐺 ∈ Mnd → ⨣ :(𝐵 × 𝐵)⟶𝐵) |
4 | simpr 488 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
5 | eqid 2798 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
6 | 1, 5 | mndidcl 17918 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵) |
7 | 6 | adantr 484 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (0g‘𝐺) ∈ 𝐵) |
8 | eqid 2798 | . . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
9 | 1, 8, 5 | mndrid 17924 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥) |
10 | 9 | eqcomd 2804 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → 𝑥 = (𝑥(+g‘𝐺)(0g‘𝐺))) |
11 | rspceov 7182 | . . . . 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 1368 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦(+g‘𝐺)𝑧)) |
13 | 1, 8, 2 | plusfval 17851 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦 ⨣ 𝑧) = (𝑦(+g‘𝐺)𝑧)) |
14 | 13 | eqeq2d 2809 | . . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥 = (𝑦 ⨣ 𝑧) ↔ 𝑥 = (𝑦(+g‘𝐺)𝑧))) |
15 | 14 | 2rexbiia 3257 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦(+g‘𝐺)𝑧)) |
16 | 12, 15 | sylibr 237 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧)) |
17 | 16 | ralrimiva 3149 | . 2 ⊢ (𝐺 ∈ Mnd → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧)) |
18 | foov 7302 | . 2 ⊢ ( ⨣ :(𝐵 × 𝐵)–onto→𝐵 ↔ ( ⨣ :(𝐵 × 𝐵)⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧))) | |
19 | 3, 17, 18 | sylanbrc 586 | 1 ⊢ (𝐺 ∈ Mnd → ⨣ :(𝐵 × 𝐵)–onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 × cxp 5517 ⟶wf 6320 –onto→wfo 6322 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 0gc0g 16705 +𝑓cplusf 17841 Mndcmnd 17903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-0g 16707 df-plusf 17843 df-mgm 17844 df-sgrp 17893 df-mnd 17904 |
This theorem is referenced by: mndfo 17927 grpplusfo 18108 |
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