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Mirrors > Home > MPE Home > Th. List > mndvlid | Structured version Visualization version GIF version |
Description: Tuple-wise left identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
mndvcl.b | ⊢ 𝐵 = (Base‘𝑀) |
mndvcl.p | ⊢ + = (+g‘𝑀) |
mndvlid.z | ⊢ 0 = (0g‘𝑀) |
Ref | Expression |
---|---|
mndvlid | ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → ((𝐼 × { 0 }) ∘f + 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapex 8421 | . . . 4 ⊢ (𝑋 ∈ (𝐵 ↑m 𝐼) → (𝐵 ∈ V ∧ 𝐼 ∈ V)) | |
2 | 1 | simprd 498 | . . 3 ⊢ (𝑋 ∈ (𝐵 ↑m 𝐼) → 𝐼 ∈ V) |
3 | 2 | adantl 484 | . 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → 𝐼 ∈ V) |
4 | elmapi 8422 | . . 3 ⊢ (𝑋 ∈ (𝐵 ↑m 𝐼) → 𝑋:𝐼⟶𝐵) | |
5 | 4 | adantl 484 | . 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → 𝑋:𝐼⟶𝐵) |
6 | mndvcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
7 | mndvlid.z | . . . 4 ⊢ 0 = (0g‘𝑀) | |
8 | 6, 7 | mndidcl 17920 | . . 3 ⊢ (𝑀 ∈ Mnd → 0 ∈ 𝐵) |
9 | 8 | adantr 483 | . 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → 0 ∈ 𝐵) |
10 | mndvcl.p | . . . 4 ⊢ + = (+g‘𝑀) | |
11 | 6, 10, 7 | mndlid 17925 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) |
12 | 11 | adantlr 713 | . 2 ⊢ (((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) |
13 | 3, 5, 9, 12 | caofid0l 7431 | 1 ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → ((𝐼 × { 0 }) ∘f + 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 {csn 4560 × cxp 5547 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ∘f cof 7401 ↑m cmap 8400 Basecbs 16477 +gcplusg 16559 0gc0g 16707 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-rep 5182 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-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-1st 7683 df-2nd 7684 df-map 8402 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 |
This theorem is referenced by: mendring 39785 |
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