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Theorem addsproplem7 27866
Description: Lemma for surreal addition properties. Putting together the three previous lemmas, we now show the second half of the inductive hypothesis unconditionally. (Contributed by Scott Fenton, 21-Jan-2025.)
Hypotheses
Ref Expression
addsproplem.1 (𝜑 → ∀𝑥 No 𝑦 No 𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
addspropord.2 (𝜑𝑋 No )
addspropord.3 (𝜑𝑌 No )
addspropord.4 (𝜑𝑍 No )
addspropord.5 (𝜑𝑌 <s 𝑍)
Assertion
Ref Expression
addsproplem7 (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Distinct variable groups:   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem addsproplem7
StepHypRef Expression
1 bdayelon 27683 . . . 4 ( bday 𝑌) ∈ On
2 fvex 6904 . . . . 5 ( bday 𝑌) ∈ V
32elon 6372 . . . 4 (( bday 𝑌) ∈ On ↔ Ord ( bday 𝑌))
41, 3mpbi 229 . . 3 Ord ( bday 𝑌)
5 bdayelon 27683 . . . 4 ( bday 𝑍) ∈ On
6 fvex 6904 . . . . 5 ( bday 𝑍) ∈ V
76elon 6372 . . . 4 (( bday 𝑍) ∈ On ↔ Ord ( bday 𝑍))
85, 7mpbi 229 . . 3 Ord ( bday 𝑍)
9 ordtri3or 6395 . . 3 ((Ord ( bday 𝑌) ∧ Ord ( bday 𝑍)) → (( bday 𝑌) ∈ ( bday 𝑍) ∨ ( bday 𝑌) = ( bday 𝑍) ∨ ( bday 𝑍) ∈ ( bday 𝑌)))
104, 8, 9mp2an 691 . 2 (( bday 𝑌) ∈ ( bday 𝑍) ∨ ( bday 𝑌) = ( bday 𝑍) ∨ ( bday 𝑍) ∈ ( bday 𝑌))
11 simpl 482 . . . . . 6 ((𝜑 ∧ ( bday 𝑌) ∈ ( bday 𝑍)) → 𝜑)
12 addsproplem.1 . . . . . 6 (𝜑 → ∀𝑥 No 𝑦 No 𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
1311, 12syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) ∈ ( bday 𝑍)) → ∀𝑥 No 𝑦 No 𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
14 addspropord.2 . . . . . 6 (𝜑𝑋 No )
1511, 14syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) ∈ ( bday 𝑍)) → 𝑋 No )
16 addspropord.3 . . . . . 6 (𝜑𝑌 No )
1711, 16syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) ∈ ( bday 𝑍)) → 𝑌 No )
18 addspropord.4 . . . . . 6 (𝜑𝑍 No )
1911, 18syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) ∈ ( bday 𝑍)) → 𝑍 No )
20 addspropord.5 . . . . . 6 (𝜑𝑌 <s 𝑍)
2111, 20syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) ∈ ( bday 𝑍)) → 𝑌 <s 𝑍)
22 simpr 484 . . . . 5 ((𝜑 ∧ ( bday 𝑌) ∈ ( bday 𝑍)) → ( bday 𝑌) ∈ ( bday 𝑍))
2313, 15, 17, 19, 21, 22addsproplem4 27863 . . . 4 ((𝜑 ∧ ( bday 𝑌) ∈ ( bday 𝑍)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
2423ex 412 . . 3 (𝜑 → (( bday 𝑌) ∈ ( bday 𝑍) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
25 simpl 482 . . . . . 6 ((𝜑 ∧ ( bday 𝑌) = ( bday 𝑍)) → 𝜑)
2625, 12syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) = ( bday 𝑍)) → ∀𝑥 No 𝑦 No 𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
2725, 14syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) = ( bday 𝑍)) → 𝑋 No )
2825, 16syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) = ( bday 𝑍)) → 𝑌 No )
2925, 18syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) = ( bday 𝑍)) → 𝑍 No )
3025, 20syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) = ( bday 𝑍)) → 𝑌 <s 𝑍)
31 simpr 484 . . . . 5 ((𝜑 ∧ ( bday 𝑌) = ( bday 𝑍)) → ( bday 𝑌) = ( bday 𝑍))
3226, 27, 28, 29, 30, 31addsproplem6 27865 . . . 4 ((𝜑 ∧ ( bday 𝑌) = ( bday 𝑍)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
3332ex 412 . . 3 (𝜑 → (( bday 𝑌) = ( bday 𝑍) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
3412adantr 480 . . . . 5 ((𝜑 ∧ ( bday 𝑍) ∈ ( bday 𝑌)) → ∀𝑥 No 𝑦 No 𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
3514adantr 480 . . . . 5 ((𝜑 ∧ ( bday 𝑍) ∈ ( bday 𝑌)) → 𝑋 No )
3616adantr 480 . . . . 5 ((𝜑 ∧ ( bday 𝑍) ∈ ( bday 𝑌)) → 𝑌 No )
3718adantr 480 . . . . 5 ((𝜑 ∧ ( bday 𝑍) ∈ ( bday 𝑌)) → 𝑍 No )
3820adantr 480 . . . . 5 ((𝜑 ∧ ( bday 𝑍) ∈ ( bday 𝑌)) → 𝑌 <s 𝑍)
39 simpr 484 . . . . 5 ((𝜑 ∧ ( bday 𝑍) ∈ ( bday 𝑌)) → ( bday 𝑍) ∈ ( bday 𝑌))
4034, 35, 36, 37, 38, 39addsproplem5 27864 . . . 4 ((𝜑 ∧ ( bday 𝑍) ∈ ( bday 𝑌)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
4140ex 412 . . 3 (𝜑 → (( bday 𝑍) ∈ ( bday 𝑌) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
4224, 33, 413jaod 1426 . 2 (𝜑 → ((( bday 𝑌) ∈ ( bday 𝑍) ∨ ( bday 𝑌) = ( bday 𝑍) ∨ ( bday 𝑍) ∈ ( bday 𝑌)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
4310, 42mpi 20 1 (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3o 1084   = wceq 1534  wcel 2099  wral 3056  cun 3942   class class class wbr 5142  Ord word 6362  Oncon0 6363  cfv 6542  (class class class)co 7414   +no cnadd 8677   No csur 27547   <s cslt 27548   bday cbday 27549   +s cadds 27850
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 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-1o 8478  df-2o 8479  df-nadd 8678  df-no 27550  df-slt 27551  df-bday 27552  df-sslt 27688  df-scut 27690  df-0s 27731  df-made 27748  df-old 27749  df-left 27751  df-right 27752  df-norec2 27840  df-adds 27851
This theorem is referenced by:  addsprop  27867
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