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Theorem mndifsplit 20659
Description: Lemma for maducoeval2 20663. (Contributed by SO, 16-Jul-2018.)
Hypotheses
Ref Expression
mndifsplit.b 𝐵 = (Base‘𝑀)
mndifsplit.0g 0 = (0g𝑀)
mndifsplit.pg + = (+g𝑀)
Assertion
Ref Expression
mndifsplit ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))

Proof of Theorem mndifsplit
StepHypRef Expression
1 pm2.21 121 . . . 4 (¬ (𝜑𝜓) → ((𝜑𝜓) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 ))))
21imp 393 . . 3 ((¬ (𝜑𝜓) ∧ (𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
323ad2antl3 1201 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
4 mndifsplit.b . . . . . 6 𝐵 = (Base‘𝑀)
5 mndifsplit.pg . . . . . 6 + = (+g𝑀)
6 mndifsplit.0g . . . . . 6 0 = (0g𝑀)
74, 5, 6mndrid 17519 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵) → (𝐴 + 0 ) = 𝐴)
873adant3 1125 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → (𝐴 + 0 ) = 𝐴)
98adantr 466 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → (𝐴 + 0 ) = 𝐴)
10 iftrue 4229 . . . . 5 (𝜑 → if(𝜑, 𝐴, 0 ) = 𝐴)
11 iffalse 4232 . . . . 5 𝜓 → if(𝜓, 𝐴, 0 ) = 0 )
1210, 11oveqan12d 6811 . . . 4 ((𝜑 ∧ ¬ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = (𝐴 + 0 ))
1312adantl 467 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = (𝐴 + 0 ))
14 iftrue 4229 . . . . 5 ((𝜑𝜓) → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
1514orcs 855 . . . 4 (𝜑 → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
1615ad2antrl 699 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
179, 13, 163eqtr4rd 2815 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
184, 5, 6mndlid 17518 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵) → ( 0 + 𝐴) = 𝐴)
19183adant3 1125 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → ( 0 + 𝐴) = 𝐴)
2019adantr 466 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → ( 0 + 𝐴) = 𝐴)
21 iffalse 4232 . . . . 5 𝜑 → if(𝜑, 𝐴, 0 ) = 0 )
22 iftrue 4229 . . . . 5 (𝜓 → if(𝜓, 𝐴, 0 ) = 𝐴)
2321, 22oveqan12d 6811 . . . 4 ((¬ 𝜑𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 𝐴))
2423adantl 467 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 𝐴))
2514olcs 856 . . . 4 (𝜓 → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
2625ad2antll 700 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
2720, 24, 263eqtr4rd 2815 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
28 simp1 1129 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → 𝑀 ∈ Mnd)
294, 6mndidcl 17515 . . . . . 6 (𝑀 ∈ Mnd → 0𝐵)
3028, 29syl 17 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → 0𝐵)
314, 5, 6mndlid 17518 . . . . 5 ((𝑀 ∈ Mnd ∧ 0𝐵) → ( 0 + 0 ) = 0 )
3228, 30, 31syl2anc 565 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → ( 0 + 0 ) = 0 )
3332adantr 466 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → ( 0 + 0 ) = 0 )
3421, 11oveqan12d 6811 . . . 4 ((¬ 𝜑 ∧ ¬ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 0 ))
3534adantl 467 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 0 ))
36 ioran 912 . . . . 5 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
37 iffalse 4232 . . . . 5 (¬ (𝜑𝜓) → if((𝜑𝜓), 𝐴, 0 ) = 0 )
3836, 37sylbir 225 . . . 4 ((¬ 𝜑 ∧ ¬ 𝜓) → if((𝜑𝜓), 𝐴, 0 ) = 0 )
3938adantl 467 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = 0 )
4033, 35, 393eqtr4rd 2815 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
413, 17, 27, 404casesdan 1026 1 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wo 826  w3a 1070   = wceq 1630  wcel 2144  ifcif 4223  cfv 6031  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  0gc0g 16307  Mndcmnd 17501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-riota 6753  df-ov 6795  df-0g 16309  df-mgm 17449  df-sgrp 17491  df-mnd 17502
This theorem is referenced by:  maducoeval2  20663  madugsum  20666
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