Theorem addsproplem6 28393

Description: Lemma for surreal addition properties. Finally, we show the second half of the induction hypothesis when 𝑌 and 𝑍 are the same age. (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)
addsproplem6.6	⊢ (𝜑 → (bday‘𝑌) = (bday‘𝑍))

Assertion

Ref	Expression
addsproplem6	⊢ (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 addsproplem6

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addspropord.3	. . . 4 ⊢ (𝜑 → 𝑌 ∈ No)
2		addspropord.4	. . . 4 ⊢ (𝜑 → 𝑍 ∈ No)
3		addsproplem6.6	. . . 4 ⊢ (𝜑 → (bday‘𝑌) = (bday‘𝑍))
4		addspropord.5	. . . 4 ⊢ (𝜑 → 𝑌 <s 𝑍)
5		nodense 28082	. . . 4 ⊢ (((𝑌 ∈ No ∧ 𝑍 ∈ No) ∧ ((bday‘𝑌) = (bday‘𝑍) ∧ 𝑌 <s 𝑍)) → ∃𝑚 ∈ No ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))
6	1, 2, 3, 4, 5	syl22anc 852	. . 3 ⊢ (𝜑 → ∃𝑚 ∈ No ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))
7		addsproplem.1	. . . . . . 7 ⊢ (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)))))
8		addspropord.2	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ No)
9	7, 8, 1	addsproplem3 28390	. . . . . 6 ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ (L‘𝑋)𝑎 = (𝑏 +s 𝑌)} ∪ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑌)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ (R‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)})))
10	9	simp1d 1160	. . . . 5 ⊢ (𝜑 → (𝑋 +s 𝑌) ∈ No)
11	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) ∈ No)
12	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 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)))))
13	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑋 ∈ No)
14		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ No)
15		unidm 4159	. . . . . . 7 ⊢ (((bday‘𝑋) +no (bday‘𝑚)) ∪ ((bday‘𝑋) +no (bday‘𝑚))) = ((bday‘𝑋) +no (bday‘𝑚))
16		simprr1 1240	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (bday‘𝑚) ∈ (bday‘𝑌))
17		bdayon 28171	. . . . . . . . . 10 ⊢ (bday‘𝑚) ∈ On
18		bdayon 28171	. . . . . . . . . 10 ⊢ (bday‘𝑌) ∈ On
19		bdayon 28171	. . . . . . . . . 10 ⊢ (bday‘𝑋) ∈ On
20		naddel2 8729	. . . . . . . . . 10 ⊢ (((bday‘𝑚) ∈ On ∧ (bday‘𝑌) ∈ On ∧ (bday‘𝑋) ∈ On) → ((bday‘𝑚) ∈ (bday‘𝑌) ↔ ((bday‘𝑋) +no (bday‘𝑚)) ∈ ((bday‘𝑋) +no (bday‘𝑌))))
21	17, 18, 19, 20	mp3an 1491	. . . . . . . . 9 ⊢ ((bday‘𝑚) ∈ (bday‘𝑌) ↔ ((bday‘𝑋) +no (bday‘𝑚)) ∈ ((bday‘𝑋) +no (bday‘𝑌)))
22	16, 21	sylib 221	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((bday‘𝑋) +no (bday‘𝑚)) ∈ ((bday‘𝑋) +no (bday‘𝑌)))
23		elun1 4183	. . . . . . . 8 ⊢ (((bday‘𝑋) +no (bday‘𝑚)) ∈ ((bday‘𝑋) +no (bday‘𝑌)) → ((bday‘𝑋) +no (bday‘𝑚)) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))))
24	22, 23	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((bday‘𝑋) +no (bday‘𝑚)) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))))
25	15, 24	eqeltrid 2868	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (((bday‘𝑋) +no (bday‘𝑚)) ∪ ((bday‘𝑋) +no (bday‘𝑚))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))))
26	12, 13, 14, 14, 25	addsproplem1 28388	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((𝑋 +s 𝑚) ∈ No ∧ (𝑚 <s 𝑚 → (𝑚 +s 𝑋) <s (𝑚 +s 𝑋))))
27	26	simpld 500	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ No)
28		uncom 4160	. . . . . . . . . . . 12 ⊢ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) = (((bday‘𝑋) +no (bday‘𝑍)) ∪ ((bday‘𝑋) +no (bday‘𝑌)))
29	28	eleq2i 2856	. . . . . . . . . . 11 ⊢ ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) ↔ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑍)) ∪ ((bday‘𝑋) +no (bday‘𝑌))))
30	29	imbi1i 352	. . . . . . . . . 10 ⊢ (((((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‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑍)) ∪ ((bday‘𝑋) +no (bday‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
31	30	ralbii 3112	. . . . . . . . 9 ⊢ (∀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)))) ↔ ∀𝑧 ∈ No ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑍)) ∪ ((bday‘𝑋) +no (bday‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
32	31	2ralbii 3141	. . . . . . . 8 ⊢ (∀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)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑍)) ∪ ((bday‘𝑋) +no (bday‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
33	7, 32	sylib 221	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑍)) ∪ ((bday‘𝑋) +no (bday‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
34	33, 8, 2	addsproplem3 28390	. . . . . 6 ⊢ (𝜑 → ((𝑋 +s 𝑍) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ (L‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)} ∧ {(𝑋 +s 𝑍)} <<s ({𝑒 ∣ ∃𝑓 ∈ (R‘𝑋)𝑒 = (𝑓 +s 𝑍)} ∪ {𝑔 ∣ ∃ℎ ∈ (R‘𝑍)𝑔 = (𝑋 +s ℎ)})))
35	34	simp1d 1160	. . . . 5 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ No)
36	35	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑍) ∈ No)
37	9	simp3d 1162	. . . . . 6 ⊢ (𝜑 → {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ (R‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)}))
38	37	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ (R‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)}))
39		ovex 7479	. . . . . . 7 ⊢ (𝑋 +s 𝑌) ∈ V
40	39	snid 4623	. . . . . 6 ⊢ (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)}
41	40	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)})
42		oldbday 28320	. . . . . . . . . . 11 ⊢ (((bday‘𝑌) ∈ On ∧ 𝑚 ∈ No) → (𝑚 ∈ (O‘(bday‘𝑌)) ↔ (bday‘𝑚) ∈ (bday‘𝑌)))
43	18, 14, 42	sylancr 599	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑚 ∈ (O‘(bday‘𝑌)) ↔ (bday‘𝑚) ∈ (bday‘𝑌)))
44	16, 43	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ (O‘(bday‘𝑌)))
45		simprr2 1241	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑌 <s 𝑚)
46		elright 28271	. . . . . . . . 9 ⊢ (𝑚 ∈ (R‘𝑌) ↔ (𝑚 ∈ (O‘(bday‘𝑌)) ∧ 𝑌 <s 𝑚))
47	44, 45, 46	sylanbrc 595	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ (R‘𝑌))
48		eqid 2763	. . . . . . . 8 ⊢ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)
49		oveq2 7454	. . . . . . . . 9 ⊢ (ℎ = 𝑚 → (𝑋 +s ℎ) = (𝑋 +s 𝑚))
50	49	rspceeqv 3602	. . . . . . . 8 ⊢ ((𝑚 ∈ (R‘𝑌) ∧ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)) → ∃ℎ ∈ (R‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
51	47, 48, 50	sylancl 598	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∃ℎ ∈ (R‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
52		ovex 7479	. . . . . . . 8 ⊢ (𝑋 +s 𝑚) ∈ V
53		eqeq1 2767	. . . . . . . . 9 ⊢ (𝑔 = (𝑋 +s 𝑚) → (𝑔 = (𝑋 +s ℎ) ↔ (𝑋 +s 𝑚) = (𝑋 +s ℎ)))
54	53	rexbidv 3190	. . . . . . . 8 ⊢ (𝑔 = (𝑋 +s 𝑚) → (∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ) ↔ ∃ℎ ∈ (R‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ)))
55	52, 54	elab 3636	. . . . . . 7 ⊢ ((𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)} ↔ ∃ℎ ∈ (R‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
56	51, 55	sylibr 237	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)})
57		elun2 4184	. . . . . 6 ⊢ ((𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)} → (𝑋 +s 𝑚) ∈ ({𝑒 ∣ ∃𝑓 ∈ (R‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)}))
58	56, 57	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ ({𝑒 ∣ ∃𝑓 ∈ (R‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)}))
59	38, 41, 58	sltssepcd 28191	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) <s (𝑋 +s 𝑚))
60	33	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑍)) ∪ ((bday‘𝑋) +no (bday‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
61	2	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑍 ∈ No)
62	60, 13, 61	addsproplem3 28390	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((𝑋 +s 𝑍) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ (L‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)} ∧ {(𝑋 +s 𝑍)} <<s ({𝑒 ∣ ∃𝑓 ∈ (R‘𝑋)𝑒 = (𝑓 +s 𝑍)} ∪ {𝑔 ∣ ∃ℎ ∈ (R‘𝑍)𝑔 = (𝑋 +s ℎ)})))
63	62	simp2d 1161	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ({𝑎 ∣ ∃𝑏 ∈ (L‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)})
64	3	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (bday‘𝑌) = (bday‘𝑍))
65	16, 64	eleqtrd 2866	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (bday‘𝑚) ∈ (bday‘𝑍))
66		bdayon 28171	. . . . . . . . . . 11 ⊢ (bday‘𝑍) ∈ On
67		oldbday 28320	. . . . . . . . . . 11 ⊢ (((bday‘𝑍) ∈ On ∧ 𝑚 ∈ No) → (𝑚 ∈ (O‘(bday‘𝑍)) ↔ (bday‘𝑚) ∈ (bday‘𝑍)))
68	66, 14, 67	sylancr 599	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑚 ∈ (O‘(bday‘𝑍)) ↔ (bday‘𝑚) ∈ (bday‘𝑍)))
69	65, 68	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ (O‘(bday‘𝑍)))
70		simprr3 1242	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 <s 𝑍)
71		elleft 28270	. . . . . . . . 9 ⊢ (𝑚 ∈ (L‘𝑍) ↔ (𝑚 ∈ (O‘(bday‘𝑍)) ∧ 𝑚 <s 𝑍))
72	69, 70, 71	sylanbrc 595	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ (L‘𝑍))
73		oveq2 7454	. . . . . . . . 9 ⊢ (𝑑 = 𝑚 → (𝑋 +s 𝑑) = (𝑋 +s 𝑚))
74	73	rspceeqv 3602	. . . . . . . 8 ⊢ ((𝑚 ∈ (L‘𝑍) ∧ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)) → ∃𝑑 ∈ (L‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
75	72, 48, 74	sylancl 598	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∃𝑑 ∈ (L‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
76		eqeq1 2767	. . . . . . . . 9 ⊢ (𝑐 = (𝑋 +s 𝑚) → (𝑐 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑚) = (𝑋 +s 𝑑)))
77	76	rexbidv 3190	. . . . . . . 8 ⊢ (𝑐 = (𝑋 +s 𝑚) → (∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ (L‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑)))
78	52, 77	elab 3636	. . . . . . 7 ⊢ ((𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑)} ↔ ∃𝑑 ∈ (L‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
79	75, 78	sylibr 237	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑)})
80		elun2 4184	. . . . . 6 ⊢ ((𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑)} → (𝑋 +s 𝑚) ∈ ({𝑎 ∣ ∃𝑏 ∈ (L‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑)}))
81	79, 80	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ ({𝑎 ∣ ∃𝑏 ∈ (L‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑)}))
82		ovex 7479	. . . . . . 7 ⊢ (𝑋 +s 𝑍) ∈ V
83	82	snid 4623	. . . . . 6 ⊢ (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)}
84	83	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)})
85	63, 81, 84	sltssepcd 28191	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) <s (𝑋 +s 𝑍))
86	11, 27, 36, 59, 85	ltstrd 28153	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
87	6, 86	rexlimddv 3173	. 2 ⊢ (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
88	1, 8	addscomd 28386	. 2 ⊢ (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
89	2, 8	addscomd 28386	. 2 ⊢ (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
90	87, 88, 89	3brtr4d 5141	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ∪ cun 4152  {csn 4588   class class class wbr 5106  Oncon0 6383  ‘cfv 6561  (class class class)co 7446   +no cnadd 8705  Nocsur 28030   <s clts 28031  bdaycbday 28032   <<s cslts 28176  Ocold 28242  Lcleft 28244  Rcright 28245   +_s cadds 28378

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-2o 8508  df-nadd 8706  df-no 28033  df-lts 28034  df-bday 28035  df-slts 28177  df-cuts 28179  df-0s 28226  df-made 28246  df-old 28247  df-left 28249  df-right 28250  df-norec2 28368  df-adds 28379

This theorem is used by:  addsproplem7  28394