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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 17132 | . 2 ⊢ (𝐺 ∈ Mnd → ⨣ :(𝐵 × 𝐵)⟶𝐵) |
4 | simpr 476 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
5 | eqid 2610 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
6 | 1, 5 | mndidcl 17131 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵) |
7 | 6 | adantr 480 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (0g‘𝐺) ∈ 𝐵) |
8 | eqid 2610 | . . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
9 | 1, 8, 5 | mndrid 17135 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥) |
10 | 9 | eqcomd 2616 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → 𝑥 = (𝑥(+g‘𝐺)(0g‘𝐺))) |
11 | rspceov 6590 | . . . . 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 1318 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦(+g‘𝐺)𝑧)) |
13 | 1, 8, 2 | plusfval 17071 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦 ⨣ 𝑧) = (𝑦(+g‘𝐺)𝑧)) |
14 | 13 | eqeq2d 2620 | . . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥 = (𝑦 ⨣ 𝑧) ↔ 𝑥 = (𝑦(+g‘𝐺)𝑧))) |
15 | 14 | 2rexbiia 3037 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦(+g‘𝐺)𝑧)) |
16 | 12, 15 | sylibr 223 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧)) |
17 | 16 | ralrimiva 2949 | . 2 ⊢ (𝐺 ∈ Mnd → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧)) |
18 | foov 6706 | . 2 ⊢ ( ⨣ :(𝐵 × 𝐵)–onto→𝐵 ↔ ( ⨣ :(𝐵 × 𝐵)⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧))) | |
19 | 3, 17, 18 | sylanbrc 695 | 1 ⊢ (𝐺 ∈ Mnd → ⨣ :(𝐵 × 𝐵)–onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 × cxp 5036 ⟶wf 5800 –onto→wfo 5802 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 +𝑓cplusf 17062 Mndcmnd 17117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-0g 15925 df-plusf 17064 df-mgm 17065 df-sgrp 17107 df-mnd 17118 |
This theorem is referenced by: mndfo 17138 grpplusfo 17258 |
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