Theorem addsproplem7 28366

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 28143	. . . 4 ⊢ (bday‘𝑌) ∈ On
2		fvex 6898	. . . . 5 ⊢ (bday‘𝑌) ∈ V
3	2	elon 6371	. . . 4 ⊢ ((bday‘𝑌) ∈ On ↔ Ord (bday‘𝑌))
4	1, 3	mpbi 233	. . 3 ⊢ Ord (bday‘𝑌)
5		bdayon 28143	. . . 4 ⊢ (bday‘𝑍) ∈ On
6		fvex 6898	. . . . 5 ⊢ (bday‘𝑍) ∈ V
7	6	elon 6371	. . . 4 ⊢ ((bday‘𝑍) ∈ On ↔ Ord (bday‘𝑍))
8	5, 7	mpbi 233	. . 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)))
10	4, 8, 9	mp2an 705	. 2 ⊢ ((bday‘𝑌) ∈ (bday‘𝑍) ∨ (bday‘𝑌) = (bday‘𝑍) ∨ (bday‘𝑍) ∈ (bday‘𝑌))
11		simpl 488	. . . . . 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 18	. . . . 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 18	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) ∈ (bday‘𝑍)) → 𝑋 ∈ No)
16		addspropord.3	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ No)
17	11, 16	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) ∈ (bday‘𝑍)) → 𝑌 ∈ No)
18		addspropord.4	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ No)
19	11, 18	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) ∈ (bday‘𝑍)) → 𝑍 ∈ No)
20		addspropord.5	. . . . . 6 ⊢ (𝜑 → 𝑌 <s 𝑍)
21	11, 20	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) ∈ (bday‘𝑍)) → 𝑌 <s 𝑍)
22		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) ∈ (bday‘𝑍)) → (bday‘𝑌) ∈ (bday‘𝑍))
23	13, 15, 17, 19, 21, 22	addsproplem4 28363	. . . 4 ⊢ ((𝜑 ∧ (bday‘𝑌) ∈ (bday‘𝑍)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
24	23	ex 418	. . 3 ⊢ (𝜑 → ((bday‘𝑌) ∈ (bday‘𝑍) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
25		simpl 488	. . . . . 6 ⊢ ((𝜑 ∧ (bday‘𝑌) = (bday‘𝑍)) → 𝜑)
26	25, 12	syl 18	. . . . 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 18	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) = (bday‘𝑍)) → 𝑋 ∈ No)
28	25, 16	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) = (bday‘𝑍)) → 𝑌 ∈ No)
29	25, 18	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) = (bday‘𝑍)) → 𝑍 ∈ No)
30	25, 20	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) = (bday‘𝑍)) → 𝑌 <s 𝑍)
31		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) = (bday‘𝑍)) → (bday‘𝑌) = (bday‘𝑍))
32	26, 27, 28, 29, 30, 31	addsproplem6 28365	. . . 4 ⊢ ((𝜑 ∧ (bday‘𝑌) = (bday‘𝑍)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
33	32	ex 418	. . 3 ⊢ (𝜑 → ((bday‘𝑌) = (bday‘𝑍) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
34	12	adantr 486	. . . . 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 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑍) ∈ (bday‘𝑌)) → 𝑋 ∈ No)
36	16	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑍) ∈ (bday‘𝑌)) → 𝑌 ∈ No)
37	18	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑍) ∈ (bday‘𝑌)) → 𝑍 ∈ No)
38	20	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑍) ∈ (bday‘𝑌)) → 𝑌 <s 𝑍)
39		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑍) ∈ (bday‘𝑌)) → (bday‘𝑍) ∈ (bday‘𝑌))
40	34, 35, 36, 37, 38, 39	addsproplem5 28364	. . . 4 ⊢ ((𝜑 ∧ (bday‘𝑍) ∈ (bday‘𝑌)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
41	40	ex 418	. . 3 ⊢ (𝜑 → ((bday‘𝑍) ∈ (bday‘𝑌) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
42	24, 33, 41	3jaod 1456	. 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 21	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ w3o 1102   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ∪ cun 4149   class class class wbr 5102  Ord word 6361  Oncon0 6362  ‘cfv 6538  (class class class)co 7420   +no cnadd 8677  Nocsur 28002   <s clts 28003  bdaycbday 28004   +_s cadds 28350

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-nadd 8678  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151  df-0s 28198  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec2 28340  df-adds 28351

This theorem is used by:  addsprop  28367