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Theorem padd4N 39222
Description: Rearrangement of 4 terms in a projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
paddass.a 𝐴 = (Atomsβ€˜πΎ)
paddass.p + = (+π‘ƒβ€˜πΎ)
Assertion
Ref Expression
padd4N ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ ((𝑋 + π‘Œ) + (𝑍 + π‘Š)) = ((𝑋 + 𝑍) + (π‘Œ + π‘Š)))
Proof of Theorem padd4N
StepHypRef Expression
1 simp1 1133 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ 𝐾 ∈ HL)
2 simp2r 1197 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ π‘Œ βŠ† 𝐴)
3 simp3l 1198 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ 𝑍 βŠ† 𝐴)
4 simp3r 1199 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ π‘Š βŠ† 𝐴)
5 paddass.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
6 paddass.p . . . . 5 + = (+π‘ƒβ€˜πΎ)
75, 6padd12N 39221 . . . 4 ((𝐾 ∈ HL ∧ (π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ (π‘Œ + (𝑍 + π‘Š)) = (𝑍 + (π‘Œ + π‘Š)))
81, 2, 3, 4, 7syl13anc 1369 . . 3 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ (π‘Œ + (𝑍 + π‘Š)) = (𝑍 + (π‘Œ + π‘Š)))
98oveq2d 7420 . 2 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ (𝑋 + (π‘Œ + (𝑍 + π‘Š))) = (𝑋 + (𝑍 + (π‘Œ + π‘Š))))
10 simp2l 1196 . . 3 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ 𝑋 βŠ† 𝐴)
115, 6paddssat 39196 . . . 4 ((𝐾 ∈ HL ∧ 𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) β†’ (𝑍 + π‘Š) βŠ† 𝐴)
121, 3, 4, 11syl3anc 1368 . . 3 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ (𝑍 + π‘Š) βŠ† 𝐴)
135, 6paddass 39220 . . 3 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ (𝑍 + π‘Š) βŠ† 𝐴)) β†’ ((𝑋 + π‘Œ) + (𝑍 + π‘Š)) = (𝑋 + (π‘Œ + (𝑍 + π‘Š))))
141, 10, 2, 12, 13syl13anc 1369 . 2 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ ((𝑋 + π‘Œ) + (𝑍 + π‘Š)) = (𝑋 + (π‘Œ + (𝑍 + π‘Š))))
155, 6paddssat 39196 . . . 4 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) β†’ (π‘Œ + π‘Š) βŠ† 𝐴)
161, 2, 4, 15syl3anc 1368 . . 3 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ (π‘Œ + π‘Š) βŠ† 𝐴)
175, 6paddass 39220 . . 3 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴 ∧ (π‘Œ + π‘Š) βŠ† 𝐴)) β†’ ((𝑋 + 𝑍) + (π‘Œ + π‘Š)) = (𝑋 + (𝑍 + (π‘Œ + π‘Š))))
181, 10, 3, 16, 17syl13anc 1369 . 2 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ ((𝑋 + 𝑍) + (π‘Œ + π‘Š)) = (𝑋 + (𝑍 + (π‘Œ + π‘Š))))
199, 14, 183eqtr4d 2776 1 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ ((𝑋 + π‘Œ) + (𝑍 + π‘Š)) = ((𝑋 + 𝑍) + (π‘Œ + π‘Š)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943  β€˜cfv 6536  (class class class)co 7404  Atomscatm 38644  HLchlt 38731  +𝑃cpadd 39177
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-rmo 3370  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-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-lat 18395  df-clat 18462  df-oposet 38557  df-ol 38559  df-oml 38560  df-covers 38647  df-ats 38648  df-atl 38679  df-cvlat 38703  df-hlat 38732  df-padd 39178
This theorem is referenced by:  paddclN  39224
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