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Theorem negsval 27490
Description: The value of the surreal negation function. (Contributed by Scott Fenton, 20-Aug-2024.)
Assertion
Ref Expression
negsval (𝐴 No → ( -us𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))

Proof of Theorem negsval
Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-negs 27486 . . 3 -us = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥)))))
21norecov 27421 . 2 (𝐴 No → ( -us𝐴) = (𝐴(𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))))( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴)))))
3 elex 3493 . . 3 (𝐴 No 𝐴 ∈ V)
4 negsfn 27488 . . . . . 6 -us Fn No
5 fnfun 6647 . . . . . 6 ( -us Fn No → Fun -us )
64, 5ax-mp 5 . . . . 5 Fun -us
7 fvex 6902 . . . . . 6 ( L ‘𝐴) ∈ V
8 fvex 6902 . . . . . 6 ( R ‘𝐴) ∈ V
97, 8unex 7730 . . . . 5 (( L ‘𝐴) ∪ ( R ‘𝐴)) ∈ V
10 resfunexg 7214 . . . . 5 ((Fun -us ∧ (( L ‘𝐴) ∪ ( R ‘𝐴)) ∈ V) → ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) ∈ V)
116, 9, 10mp2an 691 . . . 4 ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) ∈ V
1211a1i 11 . . 3 (𝐴 No → ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) ∈ V)
13 ovexd 7441 . . 3 (𝐴 No → ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))) ∈ V)
14 fveq2 6889 . . . . . 6 (𝑥 = 𝐴 → ( R ‘𝑥) = ( R ‘𝐴))
1514imaeq2d 6058 . . . . 5 (𝑥 = 𝐴 → (𝑛 “ ( R ‘𝑥)) = (𝑛 “ ( R ‘𝐴)))
16 fveq2 6889 . . . . . 6 (𝑥 = 𝐴 → ( L ‘𝑥) = ( L ‘𝐴))
1716imaeq2d 6058 . . . . 5 (𝑥 = 𝐴 → (𝑛 “ ( L ‘𝑥)) = (𝑛 “ ( L ‘𝐴)))
1815, 17oveq12d 7424 . . . 4 (𝑥 = 𝐴 → ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))) = ((𝑛 “ ( R ‘𝐴)) |s (𝑛 “ ( L ‘𝐴))))
19 imaeq1 6053 . . . . 5 (𝑛 = ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) → (𝑛 “ ( R ‘𝐴)) = (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)))
20 imaeq1 6053 . . . . 5 (𝑛 = ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) → (𝑛 “ ( L ‘𝐴)) = (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴)))
2119, 20oveq12d 7424 . . . 4 (𝑛 = ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) → ((𝑛 “ ( R ‘𝐴)) |s (𝑛 “ ( L ‘𝐴))) = ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))))
22 eqid 2733 . . . 4 (𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥)))) = (𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))))
2318, 21, 22ovmpog 7564 . . 3 ((𝐴 ∈ V ∧ ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) ∈ V ∧ ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))) ∈ V) → (𝐴(𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))))( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴)))) = ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))))
243, 12, 13, 23syl3anc 1372 . 2 (𝐴 No → (𝐴(𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))))( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴)))) = ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))))
25 ssun2 4173 . . . . 5 ( R ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴))
26 resima2 6015 . . . . 5 (( R ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴)) → (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) = ( -us “ ( R ‘𝐴)))
2725, 26ax-mp 5 . . . 4 (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) = ( -us “ ( R ‘𝐴))
28 ssun1 4172 . . . . 5 ( L ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴))
29 resima2 6015 . . . . 5 (( L ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴)) → (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴)) = ( -us “ ( L ‘𝐴)))
3028, 29ax-mp 5 . . . 4 (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴)) = ( -us “ ( L ‘𝐴))
3127, 30oveq12i 7418 . . 3 ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))
3231a1i 11 . 2 (𝐴 No → ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
332, 24, 323eqtrd 2777 1 (𝐴 No → ( -us𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3475  cun 3946  wss 3948  cres 5678  cima 5679  Fun wfun 6535   Fn wfn 6536  cfv 6541  (class class class)co 7406  cmpo 7408   No csur 27133   |s cscut 27274   L cleft 27330   R cright 27331   -us cnegs 27484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-1o 8463  df-2o 8464  df-no 27136  df-slt 27137  df-bday 27138  df-sslt 27273  df-scut 27275  df-made 27332  df-old 27333  df-left 27335  df-right 27336  df-norec 27412  df-negs 27486
This theorem is referenced by:  negs0s  27491  negsproplem3  27494  negsid  27505  negsunif  27519  negsbdaylem  27520
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