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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 2724 | . . . 4 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺) | |
3 | 1, 2 | mndpfo 18680 | . . 3 ⊢ (𝐺 ∈ Mnd → (+𝑓‘𝐺):(𝐵 × 𝐵)–onto→𝐵) |
4 | 3 | adantr 480 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → (+𝑓‘𝐺):(𝐵 × 𝐵)–onto→𝐵) |
5 | mndfo.p | . . . . . 6 ⊢ + = (+g‘𝐺) | |
6 | 1, 5, 2 | plusfeq 18571 | . . . . 5 ⊢ ( + Fn (𝐵 × 𝐵) → (+𝑓‘𝐺) = + ) |
7 | 6 | eqcomd 2730 | . . . 4 ⊢ ( + Fn (𝐵 × 𝐵) → + = (+𝑓‘𝐺)) |
8 | 7 | adantl 481 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + = (+𝑓‘𝐺)) |
9 | foeq1 6791 | . . 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 395 = wceq 1533 ∈ wcel 2098 × cxp 5664 Fn wfn 6528 –onto→wfo 6531 ‘cfv 6533 Basecbs 17143 +gcplusg 17196 +𝑓cplusf 18560 Mndcmnd 18657 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fo 6539 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-0g 17386 df-plusf 18562 df-mgm 18563 df-sgrp 18642 df-mnd 18658 |
This theorem is referenced by: (None) |
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