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Mirrors > Home > MPE Home > Th. List > mndvass | Structured version Visualization version GIF version |
Description: Tuple-wise associativity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
mndvcl.b | ⊢ 𝐵 = (Base‘𝑀) |
mndvcl.p | ⊢ + = (+g‘𝑀) |
Ref | Expression |
---|---|
mndvass | ⊢ ((𝑀 ∈ Mnd ∧ (𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼) ∧ 𝑍 ∈ (𝐵 ↑m 𝐼))) → ((𝑋 ∘f + 𝑌) ∘f + 𝑍) = (𝑋 ∘f + (𝑌 ∘f + 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapex 8841 | . . . . 5 ⊢ (𝑋 ∈ (𝐵 ↑m 𝐼) → (𝐵 ∈ V ∧ 𝐼 ∈ V)) | |
2 | 1 | simprd 495 | . . . 4 ⊢ (𝑋 ∈ (𝐵 ↑m 𝐼) → 𝐼 ∈ V) |
3 | 2 | 3ad2ant1 1130 | . . 3 ⊢ ((𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼) ∧ 𝑍 ∈ (𝐵 ↑m 𝐼)) → 𝐼 ∈ V) |
4 | 3 | adantl 481 | . 2 ⊢ ((𝑀 ∈ Mnd ∧ (𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼) ∧ 𝑍 ∈ (𝐵 ↑m 𝐼))) → 𝐼 ∈ V) |
5 | elmapi 8842 | . . . 4 ⊢ (𝑋 ∈ (𝐵 ↑m 𝐼) → 𝑋:𝐼⟶𝐵) | |
6 | 5 | 3ad2ant1 1130 | . . 3 ⊢ ((𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼) ∧ 𝑍 ∈ (𝐵 ↑m 𝐼)) → 𝑋:𝐼⟶𝐵) |
7 | 6 | adantl 481 | . 2 ⊢ ((𝑀 ∈ Mnd ∧ (𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼) ∧ 𝑍 ∈ (𝐵 ↑m 𝐼))) → 𝑋:𝐼⟶𝐵) |
8 | elmapi 8842 | . . . 4 ⊢ (𝑌 ∈ (𝐵 ↑m 𝐼) → 𝑌:𝐼⟶𝐵) | |
9 | 8 | 3ad2ant2 1131 | . . 3 ⊢ ((𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼) ∧ 𝑍 ∈ (𝐵 ↑m 𝐼)) → 𝑌:𝐼⟶𝐵) |
10 | 9 | adantl 481 | . 2 ⊢ ((𝑀 ∈ Mnd ∧ (𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼) ∧ 𝑍 ∈ (𝐵 ↑m 𝐼))) → 𝑌:𝐼⟶𝐵) |
11 | elmapi 8842 | . . . 4 ⊢ (𝑍 ∈ (𝐵 ↑m 𝐼) → 𝑍:𝐼⟶𝐵) | |
12 | 11 | 3ad2ant3 1132 | . . 3 ⊢ ((𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼) ∧ 𝑍 ∈ (𝐵 ↑m 𝐼)) → 𝑍:𝐼⟶𝐵) |
13 | 12 | adantl 481 | . 2 ⊢ ((𝑀 ∈ Mnd ∧ (𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼) ∧ 𝑍 ∈ (𝐵 ↑m 𝐼))) → 𝑍:𝐼⟶𝐵) |
14 | mndvcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
15 | mndvcl.p | . . . 4 ⊢ + = (+g‘𝑀) | |
16 | 14, 15 | mndass 18673 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
17 | 16 | adantlr 712 | . 2 ⊢ (((𝑀 ∈ Mnd ∧ (𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼) ∧ 𝑍 ∈ (𝐵 ↑m 𝐼))) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
18 | 4, 7, 10, 13, 17 | caofass 7703 | 1 ⊢ ((𝑀 ∈ Mnd ∧ (𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼) ∧ 𝑍 ∈ (𝐵 ↑m 𝐼))) → ((𝑋 ∘f + 𝑌) ∘f + 𝑍) = (𝑋 ∘f + (𝑌 ∘f + 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 ∘f cof 7664 ↑m cmap 8819 Basecbs 17150 +gcplusg 17203 Mndcmnd 18664 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-1st 7971 df-2nd 7972 df-map 8821 df-sgrp 18649 df-mnd 18665 |
This theorem is referenced by: mendring 42494 |
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