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Mirrors > Home > MPE Home > Th. List > mndfo | Structured version Visualization version GIF version |
Description: The addition operation of a monoid is an onto function (assuming it is a function). (Contributed by Mario Carneiro, 11-Oct-2013.) (Proof shortened by AV, 23-Jan-2020.) |
Ref | Expression |
---|---|
mndfo.b | ⊢ 𝐵 = (Base‘𝐺) |
mndfo.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
mndfo | ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)–onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndfo.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2821 | . . . 4 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺) | |
3 | 1, 2 | mndpfo 17928 | . . 3 ⊢ (𝐺 ∈ Mnd → (+𝑓‘𝐺):(𝐵 × 𝐵)–onto→𝐵) |
4 | 3 | adantr 483 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → (+𝑓‘𝐺):(𝐵 × 𝐵)–onto→𝐵) |
5 | mndfo.p | . . . . . 6 ⊢ + = (+g‘𝐺) | |
6 | 1, 5, 2 | plusfeq 17854 | . . . . 5 ⊢ ( + Fn (𝐵 × 𝐵) → (+𝑓‘𝐺) = + ) |
7 | 6 | eqcomd 2827 | . . . 4 ⊢ ( + Fn (𝐵 × 𝐵) → + = (+𝑓‘𝐺)) |
8 | 7 | adantl 484 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + = (+𝑓‘𝐺)) |
9 | foeq1 6580 | . . 3 ⊢ ( + = (+𝑓‘𝐺) → ( + :(𝐵 × 𝐵)–onto→𝐵 ↔ (+𝑓‘𝐺):(𝐵 × 𝐵)–onto→𝐵)) | |
10 | 8, 9 | syl 17 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → ( + :(𝐵 × 𝐵)–onto→𝐵 ↔ (+𝑓‘𝐺):(𝐵 × 𝐵)–onto→𝐵)) |
11 | 4, 10 | mpbird 259 | 1 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)–onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 × cxp 5547 Fn wfn 6344 –onto→wfo 6347 ‘cfv 6349 Basecbs 16477 +gcplusg 16559 +𝑓cplusf 17843 Mndcmnd 17905 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fo 6355 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-0g 16709 df-plusf 17845 df-mgm 17846 df-sgrp 17895 df-mnd 17906 |
This theorem is referenced by: (None) |
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