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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 2733 | . . . 4 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺) | |
3 | 1, 2 | mndpfo 18584 | . . 3 ⊢ (𝐺 ∈ Mnd → (+𝑓‘𝐺):(𝐵 × 𝐵)–onto→𝐵) |
4 | 3 | adantr 482 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → (+𝑓‘𝐺):(𝐵 × 𝐵)–onto→𝐵) |
5 | mndfo.p | . . . . . 6 ⊢ + = (+g‘𝐺) | |
6 | 1, 5, 2 | plusfeq 18510 | . . . . 5 ⊢ ( + Fn (𝐵 × 𝐵) → (+𝑓‘𝐺) = + ) |
7 | 6 | eqcomd 2739 | . . . 4 ⊢ ( + Fn (𝐵 × 𝐵) → + = (+𝑓‘𝐺)) |
8 | 7 | adantl 483 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + = (+𝑓‘𝐺)) |
9 | foeq1 6753 | . . 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 257 | 1 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)–onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 × cxp 5632 Fn wfn 6492 –onto→wfo 6495 ‘cfv 6497 Basecbs 17088 +gcplusg 17138 +𝑓cplusf 18499 Mndcmnd 18561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fo 6503 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-0g 17328 df-plusf 18501 df-mgm 18502 df-sgrp 18551 df-mnd 18562 |
This theorem is referenced by: (None) |
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