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Theorem negsval 27327
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 27323 . . 3 -us = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥)))))
21norecov 27262 . 2 (𝐴 No → ( -us𝐴) = (𝐴(𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))))( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴)))))
3 elex 3464 . . 3 (𝐴 No 𝐴 ∈ V)
4 negsfn 27325 . . . . . 6 -us Fn No
5 fnfun 6603 . . . . . 6 ( -us Fn No → Fun -us )
64, 5ax-mp 5 . . . . 5 Fun -us
7 fvex 6856 . . . . . 6 ( L ‘𝐴) ∈ V
8 fvex 6856 . . . . . 6 ( R ‘𝐴) ∈ V
97, 8unex 7681 . . . . 5 (( L ‘𝐴) ∪ ( R ‘𝐴)) ∈ V
10 resfunexg 7166 . . . . 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 7393 . . 3 (𝐴 No → ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))) ∈ V)
14 fveq2 6843 . . . . . 6 (𝑥 = 𝐴 → ( R ‘𝑥) = ( R ‘𝐴))
1514imaeq2d 6014 . . . . 5 (𝑥 = 𝐴 → (𝑛 “ ( R ‘𝑥)) = (𝑛 “ ( R ‘𝐴)))
16 fveq2 6843 . . . . . 6 (𝑥 = 𝐴 → ( L ‘𝑥) = ( L ‘𝐴))
1716imaeq2d 6014 . . . . 5 (𝑥 = 𝐴 → (𝑛 “ ( L ‘𝑥)) = (𝑛 “ ( L ‘𝐴)))
1815, 17oveq12d 7376 . . . 4 (𝑥 = 𝐴 → ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))) = ((𝑛 “ ( R ‘𝐴)) |s (𝑛 “ ( L ‘𝐴))))
19 imaeq1 6009 . . . . 5 (𝑛 = ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) → (𝑛 “ ( R ‘𝐴)) = (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)))
20 imaeq1 6009 . . . . 5 (𝑛 = ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) → (𝑛 “ ( L ‘𝐴)) = (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴)))
2119, 20oveq12d 7376 . . . 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 2737 . . . 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 7515 . . 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 4134 . . . . 5 ( R ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴))
26 resima2 5973 . . . . 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 4133 . . . . 5 ( L ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴))
29 resima2 5973 . . . . 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 7370 . . 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 2781 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 3446  cun 3909  wss 3911  cres 5636  cima 5637  Fun wfun 6491   Fn wfn 6492  cfv 6497  (class class class)co 7358  cmpo 7360   No csur 26991   |s cscut 27125   L cleft 27178   R cright 27179   -us cnegs 27321
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-1o 8413  df-2o 8414  df-no 26994  df-slt 26995  df-bday 26996  df-sslt 27124  df-scut 27126  df-made 27180  df-old 27181  df-left 27183  df-right 27184  df-norec 27253  df-negs 27323
This theorem is referenced by:  negs0s  27328  negsproplem3  27331  negsid  27342  negsunif  27353
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