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Theorem padd4N 39313
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 1134 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ 𝐾 ∈ HL)
2 simp2r 1198 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ π‘Œ βŠ† 𝐴)
3 simp3l 1199 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ 𝑍 βŠ† 𝐴)
4 simp3r 1200 . . . 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 39312 . . . 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 1370 . . 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 7436 . 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 1197 . . 3 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ 𝑋 βŠ† 𝐴)
115, 6paddssat 39287 . . . 4 ((𝐾 ∈ HL ∧ 𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) β†’ (𝑍 + π‘Š) βŠ† 𝐴)
121, 3, 4, 11syl3anc 1369 . . 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 39311 . . 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 1370 . 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 39287 . . . 4 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) β†’ (π‘Œ + π‘Š) βŠ† 𝐴)
161, 2, 4, 15syl3anc 1369 . . 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 39311 . . 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 1370 . 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 2778 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 1085   = wceq 1534   ∈ wcel 2099   βŠ† wss 3947  β€˜cfv 6548  (class class class)co 7420  Atomscatm 38735  HLchlt 38822  +𝑃cpadd 39268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-proset 18287  df-poset 18305  df-plt 18322  df-lub 18338  df-glb 18339  df-join 18340  df-meet 18341  df-p0 18417  df-lat 18424  df-clat 18491  df-oposet 38648  df-ol 38650  df-oml 38651  df-covers 38738  df-ats 38739  df-atl 38770  df-cvlat 38794  df-hlat 38823  df-padd 39269
This theorem is referenced by:  paddclN  39315
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