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Theorem negsval 28075
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 28071 . . 3 -us = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥)))))
21norecov 27998 . 2 (𝐴 No → ( -us𝐴) = (𝐴(𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))))( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴)))))
3 elex 3509 . . 3 (𝐴 No 𝐴 ∈ V)
4 negsfn 28073 . . . . . 6 -us Fn No
5 fnfun 6679 . . . . . 6 ( -us Fn No → Fun -us )
64, 5ax-mp 5 . . . . 5 Fun -us
7 fvex 6933 . . . . . 6 ( L ‘𝐴) ∈ V
8 fvex 6933 . . . . . 6 ( R ‘𝐴) ∈ V
97, 8unex 7779 . . . . 5 (( L ‘𝐴) ∪ ( R ‘𝐴)) ∈ V
10 resfunexg 7252 . . . . 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 7483 . . 3 (𝐴 No → ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))) ∈ V)
14 fveq2 6920 . . . . . 6 (𝑥 = 𝐴 → ( R ‘𝑥) = ( R ‘𝐴))
1514imaeq2d 6089 . . . . 5 (𝑥 = 𝐴 → (𝑛 “ ( R ‘𝑥)) = (𝑛 “ ( R ‘𝐴)))
16 fveq2 6920 . . . . . 6 (𝑥 = 𝐴 → ( L ‘𝑥) = ( L ‘𝐴))
1716imaeq2d 6089 . . . . 5 (𝑥 = 𝐴 → (𝑛 “ ( L ‘𝑥)) = (𝑛 “ ( L ‘𝐴)))
1815, 17oveq12d 7466 . . . 4 (𝑥 = 𝐴 → ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))) = ((𝑛 “ ( R ‘𝐴)) |s (𝑛 “ ( L ‘𝐴))))
19 imaeq1 6084 . . . . 5 (𝑛 = ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) → (𝑛 “ ( R ‘𝐴)) = (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)))
20 imaeq1 6084 . . . . 5 (𝑛 = ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) → (𝑛 “ ( L ‘𝐴)) = (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴)))
2119, 20oveq12d 7466 . . . 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 2740 . . . 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 7609 . . 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 1371 . 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 4202 . . . . 5 ( R ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴))
26 resima2 6045 . . . . 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 4201 . . . . 5 ( L ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴))
29 resima2 6045 . . . . 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 7460 . . 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 2784 1 (𝐴 No → ( -us𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2108  Vcvv 3488  cun 3974  wss 3976  cres 5702  cima 5703  Fun wfun 6567   Fn wfn 6568  cfv 6573  (class class class)co 7448  cmpo 7450   No csur 27702   |s cscut 27845   L cleft 27902   R cright 27903   -us cnegs 28069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-negs 28071
This theorem is referenced by:  negs0s  28076  negs1s  28077  negsproplem3  28080  negsid  28091  negsunif  28105  negsbdaylem  28106
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