Theorem addsproplem7 27981

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	⊢ (𝜑 → 𝑋 ∈ No )
addspropord.3	⊢ (𝜑 → 𝑌 ∈ No )
addspropord.4	⊢ (𝜑 → 𝑍 ∈ No )
addspropord.5	⊢ (𝜑 → 𝑌 <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

Step	Hyp	Ref	Expression
1		bdayon 27758	. . . 4 ⊢ ( bday ‘𝑌) ∈ On
2		fvex 6847	. . . . 5 ⊢ ( bday ‘𝑌) ∈ V
3	2	elon 6326	. . . 4 ⊢ (( bday ‘𝑌) ∈ On ↔ Ord ( bday ‘𝑌))
4	1, 3	mpbi 230	. . 3 ⊢ Ord ( bday ‘𝑌)
5		bdayon 27758	. . . 4 ⊢ ( bday ‘𝑍) ∈ On
6		fvex 6847	. . . . 5 ⊢ ( bday ‘𝑍) ∈ V
7	6	elon 6326	. . . 4 ⊢ (( bday ‘𝑍) ∈ On ↔ Ord ( bday ‘𝑍))
8	5, 7	mpbi 230	. . 3 ⊢ Ord ( bday ‘𝑍)
9		ordtri3or 6349	. . 3 ⊢ ((Ord ( bday ‘𝑌) ∧ Ord ( bday ‘𝑍)) → (( bday ‘𝑌) ∈ ( bday ‘𝑍) ∨ ( bday ‘𝑌) = ( bday ‘𝑍) ∨ ( bday ‘𝑍) ∈ ( bday ‘𝑌)))
10	4, 8, 9	mp2an 693	. 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)))))
13	11, 12	syl 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 ⊢ (𝜑 → 𝑋 ∈ No )
15	11, 14	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → 𝑋 ∈ No )
16		addspropord.3	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ No )
17	11, 16	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → 𝑌 ∈ No )
18		addspropord.4	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ No )
19	11, 18	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → 𝑍 ∈ No )
20		addspropord.5	. . . . . 6 ⊢ (𝜑 → 𝑌 <s 𝑍)
21	11, 20	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → 𝑌 <s 𝑍)
22		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → ( bday ‘𝑌) ∈ ( bday ‘𝑍))
23	13, 15, 17, 19, 21, 22	addsproplem4 27978	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
24	23	ex 412	. . 3 ⊢ (𝜑 → (( bday ‘𝑌) ∈ ( bday ‘𝑍) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
25		simpl 482	. . . . . 6 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → 𝜑)
26	25, 12	syl 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)))))
27	25, 14	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → 𝑋 ∈ No )
28	25, 16	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → 𝑌 ∈ No )
29	25, 18	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → 𝑍 ∈ No )
30	25, 20	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → 𝑌 <s 𝑍)
31		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → ( bday ‘𝑌) = ( bday ‘𝑍))
32	26, 27, 28, 29, 30, 31	addsproplem6 27980	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
33	32	ex 412	. . 3 ⊢ (𝜑 → (( bday ‘𝑌) = ( bday ‘𝑍) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
34	12	adantr 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)))))
35	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → 𝑋 ∈ No )
36	16	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → 𝑌 ∈ No )
37	18	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → 𝑍 ∈ No )
38	20	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → 𝑌 <s 𝑍)
39		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → ( bday ‘𝑍) ∈ ( bday ‘𝑌))
40	34, 35, 36, 37, 38, 39	addsproplem5 27979	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
41	40	ex 412	. . 3 ⊢ (𝜑 → (( bday ‘𝑍) ∈ ( bday ‘𝑌) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
42	24, 33, 41	3jaod 1432	. 2 ⊢ (𝜑 → ((( bday ‘𝑌) ∈ ( bday ‘𝑍) ∨ ( bday ‘𝑌) = ( bday ‘𝑍) ∨ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
43	10, 42	mpi 20	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∨ w3o 1086   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∪ cun 3888   class class class wbr 5086  Ord word 6316  Oncon0 6317  ‘cfv 6492  (class class class)co 7360   +no cnadd 8594   No csur 27617   <s clts 27618   bday cbday 27619   +_s cadds 27965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-1o 8398  df-2o 8399  df-nadd 8595  df-no 27620  df-lts 27621  df-bday 27622  df-slts 27764  df-cuts 27766  df-0s 27813  df-made 27833  df-old 27834  df-left 27836  df-right 27837  df-norec2 27955  df-adds 27966

This theorem is referenced by:  addsprop  27982