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Theorem addsproplem1 28017
Description: Lemma for surreal addition properties. To prove closure on surreal addition we need to prove that addition is compatible with order at the same time. We do this by inducting over the maximum of two natural sums of the birthdays of surreals numbers. In the final step we will loop around and use tfr3 8438 to prove this of all surreals. This first lemma just instantiates the inductive hypothesis so we do not need to do it continuously throughout the proof. (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)))))
addsproplem1.2 (𝜑𝐴 No )
addsproplem1.3 (𝜑𝐵 No )
addsproplem1.4 (𝜑𝐶 No )
addsproplem1.5 (𝜑 → ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))))
Assertion
Ref Expression
addsproplem1 (𝜑 → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑧,𝐶   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem addsproplem1
StepHypRef Expression
1 addsproplem1.2 . . 3 (𝜑𝐴 No )
2 addsproplem1.3 . . 3 (𝜑𝐵 No )
3 addsproplem1.4 . . 3 (𝜑𝐶 No )
41, 2, 33jca 1127 . 2 (𝜑 → (𝐴 No 𝐵 No 𝐶 No ))
5 addsproplem.1 . 2 (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)))))
6 addsproplem1.5 . 2 (𝜑 → ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))))
7 fveq2 6907 . . . . . . 7 (𝑥 = 𝐴 → ( bday 𝑥) = ( bday 𝐴))
87oveq1d 7446 . . . . . 6 (𝑥 = 𝐴 → (( bday 𝑥) +no ( bday 𝑦)) = (( bday 𝐴) +no ( bday 𝑦)))
97oveq1d 7446 . . . . . 6 (𝑥 = 𝐴 → (( bday 𝑥) +no ( bday 𝑧)) = (( bday 𝐴) +no ( bday 𝑧)))
108, 9uneq12d 4179 . . . . 5 (𝑥 = 𝐴 → ((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) = ((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))))
1110eleq1d 2824 . . . 4 (𝑥 = 𝐴 → (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) ↔ ((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍)))))
12 oveq1 7438 . . . . . 6 (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
1312eleq1d 2824 . . . . 5 (𝑥 = 𝐴 → ((𝑥 +s 𝑦) ∈ No ↔ (𝐴 +s 𝑦) ∈ No ))
14 oveq2 7439 . . . . . . 7 (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
15 oveq2 7439 . . . . . . 7 (𝑥 = 𝐴 → (𝑧 +s 𝑥) = (𝑧 +s 𝐴))
1614, 15breq12d 5161 . . . . . 6 (𝑥 = 𝐴 → ((𝑦 +s 𝑥) <s (𝑧 +s 𝑥) ↔ (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)))
1716imbi2d 340 . . . . 5 (𝑥 = 𝐴 → ((𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)) ↔ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴))))
1813, 17anbi12d 632 . . . 4 (𝑥 = 𝐴 → (((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))) ↔ ((𝐴 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)))))
1911, 18imbi12d 344 . . 3 (𝑥 = 𝐴 → ((((( 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)))) ↔ (((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴))))))
20 fveq2 6907 . . . . . . 7 (𝑦 = 𝐵 → ( bday 𝑦) = ( bday 𝐵))
2120oveq2d 7447 . . . . . 6 (𝑦 = 𝐵 → (( bday 𝐴) +no ( bday 𝑦)) = (( bday 𝐴) +no ( bday 𝐵)))
2221uneq1d 4177 . . . . 5 (𝑦 = 𝐵 → ((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))) = ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))))
2322eleq1d 2824 . . . 4 (𝑦 = 𝐵 → (((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) ↔ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍)))))
24 oveq2 7439 . . . . . 6 (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
2524eleq1d 2824 . . . . 5 (𝑦 = 𝐵 → ((𝐴 +s 𝑦) ∈ No ↔ (𝐴 +s 𝐵) ∈ No ))
26 breq1 5151 . . . . . 6 (𝑦 = 𝐵 → (𝑦 <s 𝑧𝐵 <s 𝑧))
27 oveq1 7438 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
2827breq1d 5158 . . . . . 6 (𝑦 = 𝐵 → ((𝑦 +s 𝐴) <s (𝑧 +s 𝐴) ↔ (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)))
2926, 28imbi12d 344 . . . . 5 (𝑦 = 𝐵 → ((𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)) ↔ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴))))
3025, 29anbi12d 632 . . . 4 (𝑦 = 𝐵 → (((𝐴 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴))) ↔ ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)))))
3123, 30imbi12d 344 . . 3 (𝑦 = 𝐵 → ((((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)))) ↔ (((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴))))))
32 fveq2 6907 . . . . . . 7 (𝑧 = 𝐶 → ( bday 𝑧) = ( bday 𝐶))
3332oveq2d 7447 . . . . . 6 (𝑧 = 𝐶 → (( bday 𝐴) +no ( bday 𝑧)) = (( bday 𝐴) +no ( bday 𝐶)))
3433uneq2d 4178 . . . . 5 (𝑧 = 𝐶 → ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))) = ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))))
3534eleq1d 2824 . . . 4 (𝑧 = 𝐶 → (((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) ↔ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍)))))
36 breq2 5152 . . . . . 6 (𝑧 = 𝐶 → (𝐵 <s 𝑧𝐵 <s 𝐶))
37 oveq1 7438 . . . . . . 7 (𝑧 = 𝐶 → (𝑧 +s 𝐴) = (𝐶 +s 𝐴))
3837breq2d 5160 . . . . . 6 (𝑧 = 𝐶 → ((𝐵 +s 𝐴) <s (𝑧 +s 𝐴) ↔ (𝐵 +s 𝐴) <s (𝐶 +s 𝐴)))
3936, 38imbi12d 344 . . . . 5 (𝑧 = 𝐶 → ((𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)) ↔ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))
4039anbi2d 630 . . . 4 (𝑧 = 𝐶 → (((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴))) ↔ ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴)))))
4135, 40imbi12d 344 . . 3 (𝑧 = 𝐶 → ((((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)))) ↔ (((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))))
4219, 31, 41rspc3v 3638 . 2 ((𝐴 No 𝐵 No 𝐶 No ) → (∀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)))) → (((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))))
434, 5, 6, 42syl3c 66 1 (𝜑 → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  cun 3961   class class class wbr 5148  cfv 6563  (class class class)co 7431   +no cnadd 8702   No csur 27699   <s cslt 27700   bday cbday 27701   +s cadds 28007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by:  addsproplem2  28018  addsproplem6  28022
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